Sometimes you may have to add zeros to the dividend the number inside the division sign. For example: or Practice: Dividing Decimals 1. Divide 8 by 0. Divide Answers: Dividing Decimals 1. In other words, means 13 divided by So do just that insert decimal points and zeros accordingly. Answers: Changing Fractions to Decimals 1. The word percent means hundredths per hundred. For example: Changing Decimals to Percents To change decimals to percents: 1.
Move the decimal point two places to the right. Insert a percent sign. For example: 0. Eliminate the percent sign. Move the decimal point two places to the left sometimes adding zeros will be necessary. Change the decimal to a percent. Drop the percent sign. Write over Remember, the word of means multiply.
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Answers: Finding Percent of a Number 1. For what, substitute the letter x; for is, substitute an equal sign; for of substitute a multiplication sign. Change percents to decimals or fractions, whichever you find easier. Then solve the equation.
Answers: Other Applications of Percent 1. Percent—Proportion Method A proportion is a statement that says that two values expressed in fraction form are equal. Since and both have values of , it can be stated that. In the example of 60 Arithmetic and Data Analysis You can use this cross-products fact in order to solve a proportion. Because the percent is the unknown, put an x over the The number 30 is next to the word is, so it goes on top of the next fraction, and 50 is next to the word of, so it goes on the bottom of the next fraction.
Practice: Percent—Proportion Method 1. What percent of 25 is 10? Answers: Percent—Proportion Method 1. Answer: 4. Note that the terms percentage rise, percentage difference, and percentage change are the same as percent change. Find the percent decrease from to What is the percent increase in rainfall from January 2. What is the percent change from 2, to 1,? Powers and Exponents An exponent is a positive or negative number or zero placed above and to the right of a quantity.
It expresses the power to which the quantity is to be raised or lowered. In 43, 3 is the exponent. The number can be simplified as follows: A few more examples: 64 Arithmetic and Data Analysis Operations with Powers and Exponents To multiply two numbers with exponents, if the base numbers are the same, simply keep the base number and add the exponents. To multiply or divide numbers with exponents, if the base numbers are different, you must simplify each number with an exponent first and then perform the operation. To add or subtract numbers with exponents, whether the base numbers are the same or different, you must simplify each number with an exponent first and then perform the indicated operation.
A number written in scientific notation is a number between 1 and 10 and multiplied by a power of Simply place the decimal point to get a number between 1 and 10 and then count the digits to the right of the decimal to get the power of Simply place the decimal point to get a number between 1 and 10 and then count the digits from the original decimal point to the new one. That is, if a number expressed in scientific notation has a positive exponent, then its value is greater than 1, and if it has a negative exponent, then it is a positive number but is less than 1.
Practice: Scientific Notation Change the following to scientific notation: 1. To square a number, just multiply it by itself the exponent would be 2. Here is a list of the first 13 perfect squares. Here is a list of the first 8 perfect cubes. Square Roots and Cube Roots Note that square and cube roots and operations with them are often included in algebra sections, and the following will be discussed further in the algebra section.
Square Roots To find the square root of a number, you want to find some number that, when multiplied by itself, gives you the original number. In other words, to find the square root of 25, you want to find the number that, when multiplied by itself, gives you The square root of 25, then, is 5. The symbol for square root is. Here is a list of the first 11 perfect whole number square roots.
Other roots are similarly defined and identified by the index given. Special note: If no sign or a positive sign is placed in front of the square root, then the positive answer is required. Only if a negative sign is in front of 72 Arithmetic and Data Analysis the square root is a negative answer required. This notation is used on most standardized exams and will be adhered to in this book. Cube Roots To find the cube root of a number, you want to find some number that, when multiplied by itself twice, gives you the original number.
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In other words, to find the cube root of 8, you want to find the number that, when multiplied by itself twice, gives you 8. Notice that the symbol for cube root is the square root sign with a small three called the index above and to the left. In square root, an index of two is understood and usually not written. Following is a list of the first five perfect whole number cube roots: Notice that the cube root of a negative number is a real number, but that the square root of a negative number is not a real number: , which is a real number Approximating Square Roots To find the square root of a number that is not an exact square, you will need to find an approximate answer by using the procedure explained here: Approximate The is between.
To check, multiply: and 6. Square roots of nonperfect squares can be approximated or looked up in tables. Answers: Approximating Square Root Problems 1. In fractions, can be reduced to. In square roots,. To simplify a square root, first factor the number under the plified to into a counting number times the largest perfect square number that will divide into the number without leaving a remainder.
Perfect square numbers are 1, 4, 9, 16, 25, 36, For example: Then take the square root of the perfect square number: and finally write as a single expression:. Remember that most square roots cannot be simplified, as they are , or. Answers: Simplifying Square Roots 1. When all outcomes are equally likely to occur, the probability of the occurrence of a given outcome can be found by using the following formula: Examples: 1.
Using the spinner below, what is the probability of spinning a 6 in one spin? Using the spinner above, what is the probability of spinning either a 3 or a 5 in one spin? Since there are two favorable outcomes out of ten possible outcomes, the probability is , or. What is the probability that both spinners below will stop on a 3 on the first spin?
What is the probability that on two consecutive rolls of a die the numbers will be 2 and then 3? Since the probability of getting a 2 on the first roll is and the probability of getting a 3 on the second roll is , and since the rolls are independent of each other, simply multiply: 5. What is the probability of tossing heads three consecutive times with a two-sided fair coin? Since each toss is independent and the probability is the probability would be: for each toss, 6.
What is the probability of rolling two dice in one toss so that they total 5? These are all the ways of tossing a total of 5 on two dice. Thus, there are four favorable outcomes, which gives the probability of throwing a five as: 7. Three green marbles, two blue marbles, and five yellow marbles are placed in a jar.
What is the probability of selecting at random a green marble on the first draw? Since there are ten marbles total possible outcomes and three green marbles favorable outcomes , the probability is. Using the equally spaced spinner above, what is the probability of spinning a 4 or greater in one spin? Using the equally spaced spinner above, what is the probability of spinning either a 2 or a 5 on one spin? What is the probability of rolling two dice in one toss so that they total 7?
What is the probability of tossing tails four consecutive times with a two-sided fair coin? What is the probability that each equally spaced spinner above will stop on a 2 on its first spin? In a regular deck of 52 cards, what is the probability of drawing a heart on the first draw?
There are 13 hearts in a deck. Answers: Probability 1. Since there are five numbers that are 4 or greater out of the eight numbers and all the numbers are equally spaced, the probability is. Since there are two favorable outcomes out of eight possible outcomes, the probability is , or.
Since each toss is independent and the probability is probability would be for each toss, the. Since the probability that the first spinner will stop on the number 2 is , and the probability that the second spinner will stop on the number 2 is , and since each event is independent of the other, simply multiply: 6. Since there are 13 favorable outcomes out of 52 possible outcomes, the probability is , or. Combinations and Permutations If there are a number of successive choices to make and the choices are independent of each other order makes no difference , the total number of possible choices combinations is the product of each of the choices at each stage.
How many possible combinations of shirts and ties are there if there are five different color shirts and three different color ties? How many ways can you arrange the letters S, T, O, P in a row? Thus, there are 24 different ways to arrange four different letters. Following is a more difficult type of combination involving permutations. In how many ways can four out of seven books be arranged on a shelf?
Notice that the order in which the books are displayed makes a difference. The symbol to denote this is P n, r , which is read as the permutations of n things taken r at a time. If, from among five people, three executives are to be selected, how many possible combinations of executives are there? Notice that the order of selection makes no difference. The symbol used to denote this situation is C n, r , which is read as the number of combinations of n things taken r at a time.
Practice: Combinations and Permutations Problems 1. How many possible outfits could Tim wear if he has three different color shirts, four different types of slacks, and two pairs of shoes? A three-digit PIN requires the use of the numbers from 0 to 9. How many different possible PINs exist? How many different ways are there to arrange three jars in a row on a shelf? There are nine horses in a race. How many different 1st-, 2nd-, and 3rd-place finishes are possible?
A coach is selecting a starting lineup for her basketball team. She must select from among nine players to get her starting lineup of five. How many possible starting lineups could she have? How many possible combinations of a, b, c, and d taken two at a time are there? Answers: Combinations and Permutations Problems 1. To find the total number of possible combinations, simply multiply the numbers together.
Since the order of the items is affected by the previous choice s , the number of different ways equals 3! The order in which players is selected does not matter; thus, use the combinations formula. Statistics Some Basics: Measures of Central Tendencies Any measure indicating a center of a distribution is called a measure of central tendency. The arithmetic mean is the most frequently used measure of central tendency.
It is generally reliable, is easy to use, and is more stable than the median. To determine the arithmetic mean, simply total the items and then divide by the number of items. What is the arithmetic mean of 0, 12, 18, 20, 31, and 45? What is the arithmetic mean of 25, 27, 27, and 27? What is the arithmetic mean of 20 and —10? Practice: Arithmetic Mean Problems 1. Find the arithmetic mean of 3, 6, and Find the arithmetic mean of 2, 8, 15, and Find the arithmetic mean of 26, 28, 36, and Find the arithmetic mean of 3, 7, —5, and — Answers: Arithmetic Mean Problems 1.
The weighted mean is, thus, For the first nine months of the year, the average monthly rainfall was 2 inches. For the last three months of that year, rainfall averaged 4 inches per month. What was the mean monthly rainfall for the entire year? What was the mean score of all ten students? Answers: Weighted Mean Problems 1. If there is an even number of items in the set, their median is the arithmetic mean of the middle two numbers. The median is easy to calculate and is not influenced by extreme measurements. Find the median of 3, 4, 6, 9, 21, 24, Find the median of 4, 5, 6, Practice: Median Problems Find the median of each group of numbers.
Mode The set, class, or classes that appear most, or whose frequency is the greatest is the mode or modal class. In order to have a mode, there must be a repetition of a data value. Mode is not greatly influenced by extreme cases but is probably the least important or least used of the three types. For example: Find the mode of 3, 4, 8, 9, 9, 2, 6, 11 The mode is 9 because it appears more often than any other number. Practice: Mode Problems Find the mode of each group of numbers.
The range depends solely on the extreme values. For example: Find the range of the following numbers. Practice: Range Problems 1. Find the range of 2, 45, , 99 2. Find the range of 6, , , —5 Answers: Range Problems 1. A small standard deviation indicates that the data values tend to be very close to the mean value. Each colored band 88 Arithmetic and Data Analysis has a width of one standard deviation. You will find approximately At three standard deviations from the mean, approximately The basic method for calculating the standard deviation for a population is lengthy and time consuming.
It involves five steps: 1. Find the mean value for the set of data. For each data value, find the difference between it and the mean value; then square that difference. Find the sum of the squares found in Step 2. Divide the sum found in Step 3 by how many data values there are. Find the square root of the value found in Step 4. The result found in Step 4 is referred to as the variance. The square root of the variance is the standard deviation.
For example: Find the variance and standard deviation for the following set of data. Find the mean value: 2. Find the squares of the differences between the data values and the mean.
Find the sum of the squares from Step 2: Divide the sum from Step 3 by how many data values there are:. Find the square root of the value found in Step 4: standard deviation. The statistical values that would be affected are the mean, median, and mode. The statistical values that would not be affected are the range, variance, and standard deviation.
The mean, median, and mode would each increase by the amount that each data value was increased. For example: A scientist discovered that the instrument used for an experiment was off by 2 milligrams. If each weight in his experiment needed to be increased by 2 milligrams, then which of the following statistical measures would not be affected? Only range and standard deviation would not be affected. Number Sequences Progressions of numbers are sequences with some patterns. Unless the sequence has a simple repeat pattern 1, 2, 4, 1, 2, 4,.
Practice: Number Sequence Problems Find the next number in each sequence. Weight 16 ounces oz. A kilometer is about 0. A kilogram is about 2. A liter is slightly more than a quart. Converting Units of Measure Examples: 1. If 36 inches equals 1 yard, then 3 yards equals how many inches? Change 3 decades into weeks. Since 1 decade equals 10 years and 1 year equals 52 weeks, then 3 decades equal 30 years.
If 1, yards equal 1 mile, how many yards are in 5 miles? If 1 kilometer equals approximately 0. How many cups are in 3 gallons? How many ounces are in 6 pounds? If 1 kilometer equals 1, meters and 1 decameter equals 10 meters, how many decameters are in a. The numbers 1, 2, 3, 4,. The numbers 0, 1, 2, 3,. The numbers. Give the symbol or symbols for each of the following. Show three of them. List the properties that are represented by each of the following. In the number ,, which digit is in the ten thousands place? Express in expanded notation.
Round off 7. Complete the number line below: A The number 8, is divisible by which numbers between 1 and 10? Change to a mixed number. Change to an improper fraction. Change to twelfths. List all the factors of Find the greatest common factor of 18 and List the first four multiples of 7. Find the least common multiple of 6 and 8. Change 0. Change to a percent. What is the percent increase from to ? Express , in scientific notation. Approximate Using the equally spaced spinner below , what is the probability of spinning either an A or a B in one spin? What is the probability that equally spaced spinner A will stop on 5 and equally spaced spinner B will stop on 2 if each spinner is given one spin?
What is the probability of rolling two dice on one toss so that the sum is 6? What is the probability of tossing tails five consecutive times with a two-sided fair coin? How many different ways are there to arrange four books in a row on a shelf? If among five people, two game-show contestants must be chosen, how many possible combinations of contestants are there?
Find the arithmetic mean, mode, median, and range of the following set of numbers. Find the next number in the sequence. How many ounces in 12 pounds? If 1 decimeter is 10 centimeters and 1 centimeter is 10 millimeters, then how many millimeters are there in a decimeter? In how many different ways can six of nine books be arranged on a shelf? Answers Arithmetic Page numbers following each answer refer to the review section applicable to this problem type. Any number plus its additive inverse equals 0. Only true for multiplication and addition. Also used to represent a set.
For example, the cube root of is 5. The cube root symbol is ,. Expresses the power to which the quantity is to be raised or lowered. For example, 6 is a factor of 24, but 7 is not a factor of Consists of a numerator and a denominator—for example, or. For example, 0. Any number added to 0 gives the original number. Any number multiplied by 1 gives the original number. For example, and are improper fractions. If you are asked to invert , the correct answer is. It is the least common multiple of the denominators. If the group has an even number of items, the median is the average of the two middle terms.
Any nonzero number multiplied by its multiplicative inverse equals 1. The line may be thought of as an infinitely long ruler with negative numbers to the left of zero and positive numbers to the right of zero. One number follows another in some defined manner. This includes, but is not limited to, addition, subtraction, multiplication, division, raising to exponents, finding roots, and finding absolute value.
For example, multiplication takes precedence is performed before addition. For example, 5 is to 4 as 10 is to 8, or. May be written x:y, , or x is to y. For example, is the reciprocal of. For example, reduced to. A method of approximating. Used for writing very large or very small numbers—for example, 2. For example, 5 is the square root of Its symbol is. It is calculated by taking the positive square root of the variance. The variance of a population is calculated by taking each data value, subtracting it from the mean, squaring each difference, adding these squares together, and then dividing the sum by the number of data values.
Express algebraically: five increased by three times x. Solve for y: Factor completely: 8x3 — 12x2. Factor: 16a2 — Solve for y:. Factor: x2 — 2x — Solve: m2 — 2mn — 3n2. Solve for x: Solve for x:. Give the coordinates represented by points A and B. Algebra Answers Page numbers following each answer refer to the review section applicable to this problem type. The letters or variables are merely substitutes for numbers.
Initially, algebra referred to equation solving, but now it encompasses the language of algebra and the patterns of reasoning. The rules for algebra are basically the same as the rules for arithmetic. Some Basic Language Understood Multiplication When two or more letters, or a number and letter s are written next to each other, they are understood to be multiplied. Thus, 8x means 8 times x. Or ab means a times b. Or 18ab means 18 times a times b. Parentheses also represent multiplication. Thus, 3 4 means 3 times 4. Letters to Be Aware of Although they may appear in some texts, we recommend that you never use o, e, or i as variables.
Technically, e and i stand for constants or predetermined numbers, and o is too easily confused with zero. When using z, you may want to write it as —z so it is not confused with 2. Special Sets A subset is a set within a set: The set of 2, 3 is a subset of the set of 1, 2, 3. The universal set is the general category set, or the set of all those elements under consideration. The union of sets with members 1, 2, 3 and 3, 4, 5 is the set with members 1, 2, 3, 4, 5. The intersection of two or more sets is the set of elements that they share, where they intersect, or overlap.
The intersection of a set with members 1, 2, 3 and a set with members 3, 4, 5 is a set with only member 3. True or false: 2. True or false: 5. Answers: Set Theory Problems 1. True 2. True 3. True 4. False 5. Often a letter is used to stand for a number.
Variables are used to change verbal expressions into algebraic expressions. Evaluating Expressions To evaluate an expression, just replace the unknowns with grouping symbols, insert the value for the unknowns and do the arithmetic. Answers: Evaluating Expressions Problems 1. Thus, if you do the same thing to both sides of the equal sign say, add 5 to each side , the equation will still be balanced. Examples: 1. Sometimes you may have to use more than one step to solve for an unknown. To check, substitute your answer into the original equation. Add 6 to each side: Multiply each side by.
First add X to both sides. Then divide both sides by P: Operations opposite to those in the original equation were used to isolate Q. Solve for x: To solve this equation quickly, you cross-multiply. Solve for x: Multiply both sides by k: In this problem, there was no need to cross-multiply. Solve for z: 2. Solve for q: 3.
Solve for c: 4. Solve for c: Answers: Literal Equations Problems 1. Proportions Proportions are written as two ratios fractions equal to each other. Solve this proportion for x: p is to q as x is to y. First, the proportion may be rewritten: Now simply multiply each side by y. Solve this proportion for t: s is to t as r is to q. Answers : Solving Proportions for Value Problems 1. One method is: a. Multiply one or both equations by some number to make the number in front of one of the letters unknowns the same in each equation. Add or subtract the two equations to eliminate one letter.
Solve for the other unknown. Insert the value of the first unknown in one of the original equations to solve for the second unknown. Simply add or subtract. In this instance, the system does not have a unique solution. Any replacements for a and b that make one of the sentences true, will also make the other sentence true. In this situation, the system has an infinite number of solutions for a and b.
A term is a numerical or literal expression with its own sign. For instance, 9x, 4a2, and 3mpxz2 are all monomials. A polynomial consists of two or more terms. A binomial is a polynomial that consists of exactly two terms. A trinomial is a polynomial that consists of exactly three terms.
The number in front of the variable is called the coefficient. In 9y, 9 is the coefficient. Polynomials are usually arranged in one of two ways. Descending order is basically when the power of a term decreases for each succeeding term. Descending order is more commonly used. Adding and Subtracting Monomials To add or subtract monomials, follow the same rules as with signed numbers p.
Notice that you add or subtract the coefficients only and leave the variables the same. To multiply monomials, add the exponents of the same bases. When monomials are being raised to a power, the answer is obtained by multiplying the exponents of each part of the monomial by the power to which it is being raised. You can simplify the numerator first: Or, since the numerator is all multiplication, we can cancel. Practice: Dividing Monomials Problems 1. Answers: Dividing Monomials Problems 1. Adding and Subtracting Polynomials To add or subtract polynomials, just arrange like terms in columns and then add or subtract.
Or simply add or subtract like terms when rearrangement is not necessary. Add: 2. Then simplify if necessary.
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Practice: Dividing Polynomials by Monomials Problems 1. Remember: Divide by the first term, multiply, subtract, bring down. Practice: Dividing Polynomials by Polynomials Problems 1. Factoring out a Common Factor To factor out a common factor: a Find the largest common monomial factor of each term. The second factor will be a polynomial.
Find the square root of the first term and the square root of the second term. Express your answer as the product of: the sum of the quantities from step a, times the difference of those quantities.
First, factor out the greatest common factor of 2; then recognize the difference of squares t2 — Check to see if you can monomial factor factor out common terms. Place these factors in the left sides of the parentheses. For example, x x. Factor the last term and place the factors in the right sides of the parentheses.
To decide on the signs of the numbers do the following: If the sign of the last term is negative: 1. Find two numbers whose product is the last term and whose difference is the coefficient number in front of the middle term. Give the larger of these two numbers the sign of the middle term and the opposite sign to the other factor.
Find two numbers whose product is the last term and whose sum is the coefficient of the middle term. Give both factors the sign of the middle term. First, check to see if you can monomial factor factor out common terms. Since this is not possible, use double parentheses and factor the first term as follows: x x.
Multiply means inner terms and extremes outer terms to check. The next example is this type of problem. Practice: Factoring Polynomials Problems Factor each of the following. To solve a quadratic equation using factoring: 1. Put all terms on one side of the equal sign, leaving zero on the other side.
Set each factor equal to zero. Solve each of these equations. Check by inserting your answer in the original equation. Since division by 0 is impossible, variables in the denominator have certain restrictions. The denominator can never equal 0. Reduce: or 2. Reduce: 3. Reduce: Warning: Do not cancel through an addition or subtraction sign. Multiplying Algebraic Fractions To multiply algebraic fractions, first factor the numerators and denominators that are polynomials; then cancel where possible. Multiply the remaining numerators together and denominators together.
Practice: Multiplying Algebraic Fractions Problems 1. Answers: Multiplying Algebraic Fractions Problems 1. Dividing Algebraic Fractions To divide algebraic fractions, invert the fraction doing the dividing and multiply. Remember: You can cancel only after you invert. Practice: Dividing Algebraic Fractions Problems 1.
Answers: Dividing Algebraic Fractions Problems 1. Adding or Subtracting Algebraic Fractions To add or subtract algebraic fractions having a common denominator, simply keep the denominator and combine add or subtract the numerators. To add or subtract algebraic fractions having different denominators, first find a lowest common denominator LCD , change each fraction to an equivalent fraction with the common denominator, then combine each numerator. The point on this line associated with each number is called the graph of the number. Notice that number lines are spaced equally or proportionately.
A dot is used if the number is included. A hollow dot is used if the number is not included. Graph the set of x such that x This ray is often called an open ray or an open half line. The hollow dot distinguishes an open ray from a ray. Intervals An interval consists of all the numbers that lie within two certain boundaries. If the two boundaries, or fixed numbers, are included, then the interval is called a closed interval. If the fixed numbers are not included, then the interval is called an open interval.
If the interval includes only one of the boundaries, then it is called a half-open interval. The absolute value of x is written. The absolute value of a number is always positive except when the number is 0. Solve for x. There is no solution because the absolute value of any number is never negative. The answer is all real numbers, because the absolute value of any number is always positive or zero. Practice: Absolute Value Problems 1. Answers: Absolute Value Problems 1. No solution Any real number except 4 Analytic Geometry Coordinate Graphs Each point on a number line is assigned a number.
In the same way, each point in a plane is assigned a pair of numbers. These numbers represent the placement of the point relative to two intersecting lines. In coordinate graphs, two perpendicular number lines are used and are called coordinate axes. One axis is horizontal and is called the x-axis. The other is vertical Algebra and is called the y-axis. The point of intersection of the two number lines is called the origin and is represented by the coordinates 0,0. Some coordinates are noted below.
On the y-axis, numbers above 0 are positive and below 0 are negative. The x-coordinate shows the right or left direction, and the y-coordinate shows the up or down direction. These quadrants are labeled below. In quadrant II, x is always negative and y is always positive. In quadrant III, x and y are both always negative. In quadrant IV, x is always positive and y is always negative. Repeat this process to find other solutions. When giving a value for one variable, start with 0, then try 1, and so on.
Then graph the solutions. Using a simple chart is helpful. Equations whose solution sets form a straight line are called linear equations. Equations that have a variable raised to a power, show division by a variable, involve variables with square roots, or have variables multiplied together will not form a straight line when their solutions are graphed. These are called nonlinear equations. The more points plotted, the easier it is to see and describe the solution set.
Practice: Graphing Equations on the Coordinate Plane Problems State whether the following equations are linear or nonlinear. One involves the slope of the Part I: Basic Skills Review line, and the other involves the point of intersection of the line with the y-axis, known as the y-intercept.
Example: Rewrite each of the following linear equations in slope-intercept form and identify the slope value and the y-intercept location. Also, graph each linear equation. Example: Find the slope of the line passing through the points —2,6 and 3,5. Write the equation in slope-intercept form. Locate the y-intercept on the graph. This is one point on the line.
Write the slope as a ratio fraction to use to locate other points on the line. Draw the line through the points. Find the x-intercept by replacing the y-variable with the value 0; then solve for x. The x-intercept of a graph occurs when the graph is on the x-axis. When a point is on the x-axis, its y-coordinate there is 0. Find the y-intercept by replacing the x-variable with the value 0; then solve for y. The y-intercept of a graph occurs when the graph is on the y-axis. When a point is on the y-axis, its x-coordinate there is 0.
Draw a line passing through the x- and y-intercepts. Answer: Find the x-intercept by replacing y with 0 and solving for x. Find the slope, m either it is given or you need to calculate it from two given points. Find an equation of the line with a slope of —4 and a y-intercept of 3. Find an equation for the line passing through 6,4 with a slope of —3. Substitute —3 for m and use the point 6,4 and substitute 6 for x and 4 for y in order to find b. Find an equation for a line that passes through the points 5,—4 and 3,—2. Find the slope using the two points. Substitute —1 for m and use one of the two points.
You can use either 5,—4 or 3,—2. For this example, use 5,—4. Substitute 5 for x and —4 for y in order to find b. Using the point 3,—2 , substitute 3 for x and —2 for y in order to find b. The symbol is called a radical sign and is used to designate square root. To designate cube root, a small three is placed above the radical sign:. When two radical signs are next to each other, they automatically mean that the two are multiplied. The multiplication sign may be omitted. Note that the square root of a negative number is not possible within the real number system.
A completely different system of imaginary numbers is used for roots of negative numbers. The typical standardized exams use only the real number system as general practice. The so-called imaginary , numbers are multiples of the imaginary unit i: , , and so on. The ACT and some college placement exams do deal with simple imaginary numbers and their operations.
Simplifying Square Roots A square root with no sign indicated to its left represents a positive value. A square root with a negative to its left represents a negative value. In problems 1 through 3, simplify each square root expression. If each variable is nonnegative not a negative number , 5. If each variable is nonnegative, If each variable is nonnegative, 7. If each variable is nonnegative,. If x is nonnegative,. All variables are nonnegative. Answers: Simplifying Square Root Problems 1. Operations with Square Roots You may perform operations under a single radical sign.
You can add or subtract square roots themselves only if the values under the radical sign are equal. Then simply add or subtract the coefficients numbers in front of the radical sign and keep the original number in the radical sign. Note that 1 is understood in in other words,. Algebra You may not add or subtract different square roots. Always simplify if possible. These cannot be added until is simplified. Now, since both are alike under the radical sign 2.
Try to simplify each one. Always simplify the answer when possible. If x is nonnegative, 3. If each variable is nonnegative, 4. See other items More See all. Item Information Condition:. The item you've selected was not added to your cart. Add to Watchlist Unwatch. Watch list is full. Longtime member. Does not ship to Germany See details. Item location:. Hillsboro, Oregon, United States. Ships to:.
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Item specifics Condition: Acceptable: A book with obvious wear. May have some damage to the cover but integrity still intact. The binding may be slightly damaged but integrity is still intact. Possible writing in margins, possible underlining and highlighting of text, but no missing pages or anything that would compromise the legibility or understanding of the text. See all condition definitions - opens in a new window or tab Read more about the condition.
About this product. Goodwill Bks goodwillbks Shipping and handling. The seller has not specified a shipping method to Germany. Contact the seller - opens in a new window or tab and request shipping to your location. Shipping cost cannot be calculated. Please enter a valid ZIP Code. Shipping to: United States. No additional import charges at delivery! This item will be shipped through the Global Shipping Program and includes international tracking.
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