Page of Review of Radical Expressions and Equations Skills involving radicals can be divided into the following groups: Evaluate square roots or higher order roots. Simplify radical expressions. Rationalize denominators. Find domain of a radical function. Evaluate expressions involving rational exponents. Solve a radical equation. Evaluate square roots or cube roots. Example: Evaluate 9 This is a square root. The number under the radical is 9. (An asterisk * means multiply.) 9 Find a number such that the number squared is 9. This number is because 9 Example: Evaluate 8 This is a cube root. The number under the radical is 8. 8 Find a number such that the number cubed is 8. This number is because 8.
Page of Example: Evaluate 400 This is a square root. The number under the radical is 400. 400 does not have a real number solution. Find a number such that the number squared is 400. Note that 0 0 400 and 0 0 400 (not 400). Any real number squared is a positive number or zero. Example: Evaluate 400 This is a square root. The number under the radical is 400. 400 0 0. Find a number such that the number squared is 400. ( 0 0 400) Thus 0 0 Don t forget the negative sign in front of. Example: Evaluate 7 This is a cube root. The number under the radical is 7. ( ) 7 Find a number such that the number cubed will give you 7. The number is because ( ) ( ) ( ) ( ) 7.
Page of Simplify radical expressions and Rationalize denominators Usually to simplify means to rewrite the expression in such a way that it has as few radicals as possible, and that the expression under each radical does not contain perfect powers. Some rules for radicals are illustrated below. a. or ( ) b. *0 0 c. 5 + 6 5 7 5 d. 5 * 7 5*7 5 e. 5 7 5 7 5 7 5 *5 0 Also, ( 7 ) 7 7 and 5 (+ 5) 6 *6 6 6 7 Keep in mind that: 5 + 7 does not equal 5 + 7, and 5 5 does not equal. 7 7 Example: Simplify 8. The only thing which can be simplified is 8 under the square root. Can it be factored in such a way that one of the multiples is a perfect square? 8 4*7 *7 * 7 7 Factor 8 so one factor is a perfect square. The radical of a product equals the product of the radicals. Simplify the perfect square.
Page 4 of Example: Simplify 7x. Assume x 0. The expression under the square root has a perfect square. 7x 7 x 7 x The radical of a product equals the product of the radicals. Simplify the perfect square. Example: Simplify 5 k. Assume k 0. The radical is a square root. The expression under the square root is not a perfect square. 5 4 k k k 4 k k k k. Factor so there are perfect squares. The radical of a product equals the product of the radicals. Simplify the perfect square. k remains under the radical since the exponent on k is which is smaller than the index () of the radical Example: Simplify 7 8 k. Assume k 0 and q 0. 0 q The radical is a square root. The expression under the square root is not a perfect square. 7 8 7 8 0kq 0 k q 5 64 k k q 4 8 5 6 8 5 64 k k q 5 8 k k q kq k 4 6 8 Factor so there are perfect squares. The radical of a product equals the product of the radicals. Simplify the perfect squares. The radical is simplified because there are no perfect squares left under the radical and the remaining variables and numbers have exponents of one.
Page 5 of Example: Simplify 7k 5. Assume k 0. This is a cube root. Each factor is a perfect cube. and k. 5 5 7 ( k ) 7k 5 5 The radical of a product equals the product of k 7 5 k 5 ( k ) the radicals. Write each factor as a perfect cube. Simplify the cube roots. Example: Simplify 4 5 64x y. Assume x 0 and y 0. The radical is a cube root. The expression under the cube root is not a perfect cube. 64x y 4 5 4 5 64 x y The radical of a product is the product of the 4 4 x xy xy x x x x y x y y y y y radicals. Factor so there are perfect cubes. Simplify the perfect cubes. Rearrange the factors. Generally the radical factor is written last The radical is simplified because there are no perfect cubes left in the radical.
Page 6 of Example: Simplify t 8t. Assume t 0. The expression is the product of two square roots. t 8t t 8t The product of two radicals equals the radical of their product. 4t Multiply. 4 t The radical of the product equals the product of the radicals. 6 t Factor so there are perfect squares. Simplify the square roots and rearrange. t 6 Example: Simplify y 4 y + 5. Assume y. The expression is the product of two square roots. y 4 y 4 The product of two radicals equals the radical of the product. y + 5 y + 5 ( y ) 4 ( y + 5) ( y ) y + 5 y y + 5 Multiply. Factor the perfect square in the numerator. Distribute in the denominator. Simplify the perfect square.
Page 7 of Example: Simplify 6. 6 The expression is the quotient of two cube roots. 6 6 6 6 7 The quotient of two radicals equals the radical of their quotient. Divide. Simplify the perfect cube. 7 Example: Simplify 4 5 + 8 5. The expression is the sum of two radicals. The radicals are like radicals. Both contain 5. 4 5 + 8 5 (4 + 8) 5 Add the like radicals. 5 Example: Simplify 4 5 + + 9 5 + 4. The expression contains the sum of four radical terms. The like radicals can be added. 4 5 + + 9 5 + 4 4 5 + 9 5 + + 4 Rearrange so like radicals are together. ( 4 + 9) 5 + ( + ) Combine the like radicals. 5 5 + 7
Page 8 of Example: Simplify 5 7 + 9 6. The expression contains the sum of two square roots. 5 7 + 9 6 5 7 + 9 9 7 Simplify the radicals to determine if there are like radicals. 5 7 + 9 7 5 7 + 7 7 Combine the like radicals. 7 Example: Simplify x 8 + x 8. Assume x 8. The expression contains the sum of two terms containing radicals. x 8 + x 8 4 x 8 The radicals in both terms are like radicals ( x 8 ), so the terms can be added. Example: Simplify x + + 9x + 8. Assume x. The expression contains the sum of two terms. x + + 9x + 8 x + + 9( x + ) Simplify the radicals. x + + x + + x + 4 x + ( x + ) Combine the like radicals.
Page 9 of Example: Rationalize the denominator. 5 Rationalize the denominator means to find an equivalent fraction whose denominator does not contain a radical. 5 5 5 ( 5) 5 5 5 5 Multiply the numerator and the denominator by 5. (Note: You can t just square the numerator and the denominator. It will change the value of the fraction) Simplify the denominator.
Page 0 of Example: Rationalize denominator 5 5 +. The denominator of this expression is irrational because it includes an irrational number ( ). Rationalize denominator means to find an equivalent expression, but with a rational denominator. 5 (5 )(5 ) 5+ (5+ )(5 ) 5 5 + ( ) 5 5 0 + 7 0 There is a binomial in the denominator. Multiply the numerator and denominator by ( 5 ). (The conjugate of the denominator.) Find the product of the numerator and denominator. Combine like radicals. Note: Choosing ( 5 ) uses the formula ( a b)( a b) a b and b. Squaring The fraction would change the value of the fraction. +. In this example, a 5
Page of Find the domain of a function Domain The domain is a list or set of all possible inputs that yield a real number output. There are three operations we can t do with real numbers in algebra. Each of these restrict the domain. Can t divide by zero. Can t take the square root (or any even-index radical) of a negative number. Can t take the logarithm of zero or a negative number. Two common notations to write the domain are set-builder and interval notation.. Set-builder notation: Sets are typically written in braces { }. The notation is { independent variable some property or restriction about independent variable } where the vertical line is read such that. Example: All real numbers, x, less than. { xx< } Example: All real numbers, n 4 n< 6 { } n, greater than or equal to 4 and less than 6.. Interval notation: Parenthesis indicate the starting or ending value is not included and a square bracket indicates the starting or ending value is included. Within the parentheses or square bracket, we indicate the smallest value of x followed by a comma and then the largest value of x. The examples above are shown using interval notation. Example: All real numbers, x, less than., ( ) Example: All real numbers, 4,6 [ ) n, greater than or equal to 4 and less than 6.
Page of Find the domain of a radical function Example: Find the domain of y x. The function contains a square root. The expression under the square root, x, must be greater or equal to zero. x 0 x x 6 In interval notation the answer is [ 6, ) Isolate the term with a variable. Divide both sides by. ( is positive, so don t change the inequality sign) Example: Find the domain of y 5 t. The function contains a square root. The expression under the square root, 5 t, must be greater or equal to zero. 5 t 0 t 5 5 t In interval notation the answer is 5,. Isolate the term with a variable. Divide both sides by. (Remember when you multiply or divide an inequality by a negative number, the inequality sign changes direction.)
Page of Example: Find the domain of y x + 4. The function contains cube root. The expression under the cube root can be any real number. x + 4 x + 4 can be any real number ( ) (, ) Write the expression under the radical. (The expression is called the radicand.) Since the radical is a cube root the expression can be any real number. Write the domain using interval notation. Evaluate expressions involving rational exponents For the problems in this group, an expression containing rational exponents should be written using radical notation, and an expression containing radical notation should be written using rational exponents. n n The definition of rational exponent is x x n The definition of a negative exponent is x. n x All rules for exponents apply to rational and negative exponents. Often used rules are listed below. Assume: a 0 m n m n Product Rule: a a a + m Power Rule: ( a ) n m a a mn m n Quotient Rule: a n a Example: Write 7 using rational exponents. The expression contains a cube root, which could be rewritten using rational exponents. 7 7 ( ) 7 7 * Rewrite the radical using a rational exponent. Use the power rule.
Page 4 of Example: Evaluate 8. The exponent is, which is a rational number. Use the definition for rational exponents. 8 8 (8 ) ( 8) () 4 Use the power rule. The definition of a rational exponent is used. Simplify the cube root. Note: If you use a calculator, remember to use parentheses. Enter 8^(/), not 8^/.
Page 5 of Example: Evaluate 9 The exponent is, which is a negative rational number. 9 9 9 ( 9 ) ( 9 ) ( ) 7 Use the definition of a negative exponent. Use the power rule. Write the power as square root. Simplify the square root and raise the result to the third power. If you use a calculator, remember to use parentheses. Enter 9^(-/), not 9^-/
Page 6 of Solve radical equations A radical equation is an equation containing one or more radical terms. For example, x x is a radical equation. To solve means to determine all the real values which, when substituted in the equation for x, will make the statement true. All such real values should be included in the answer. Note that 0 0 is true. Hence, x 0 is a solution of the equation x x. It turns out that x is also a solution of x x verify that 4 4 4. This second solution is often ignored. To find all the solutions, follow the steps for solving radical equations given below. For equations containing one radical, the steps are:. Isolate the radical.. Square both sides of the equation. Solve the resulting equation which no longer contains radicals. This equation is often linear, quadratic, or rational. 4. Check the answers. (It is possible that some x values may be in the solution set of the resulting equation, but will not make the original equation true.) For equations with two radicals:. Isolate one of the radicals.. Square both sides of the equation.. Combine like terms. 4. Now the equation either has no radicals, or just one radical term. a. If there are no radicals follow steps and 4 under equations containing one radical. b. If there is one radical follow steps through 4 under equations containing one radical.
Page 7 of Example: Solve the equation x + 6 This is an equation (contains an equal sign). Use the steps for equations containing one radical. x + 6 ( x + ) 6 x + 6 x + 6 x 4 The radical is isolated. Square both sides. Solve the resulting linear equation. Check 4 + 6 6 6 6 6 Substitute x 4 into the original equation and simplify the results. The statement is true so the solution is x 4. Example: Solve the equation x + 6 This is an equation (contains an equal sign). Use the steps for equations containing one radical. x + 6 ( x + ) ( 6) x + 6 x 4 The radical is isolated. Square both sides. Solve the resulting linear equation. Check 4 + 6 6 6 6 6 Substitute x 4 into the original equation and simplify the results. The statement is false, so x 4 is not a solution. Note: The square root of any real number can t be negative. Hence, x + 6 can t be true.
Page 8 of Example: Solve the equation x + 4 x 40. This is an equation (contains an equal sign). It contains two radical terms. x + 4 x 40 5 x 40 x 8 ( ) x 8 x 64 65 x.5 Add like radicals. Follow the steps for equations containing one radical. Divide both sides by 5. The radical is isolated. Square both sides. Solve the resulting linear equation. Check 65 65 ( ) ( ) + 4 40 65 + 4 65 40 64 + 4 64 40 ( ) 8+ 4 8 40 8+ 40 40 40 65 Substitute x into the original equation and simplify the results. 65 The statement is true so x is a solution to the equation.
Page 9 of Example: Solve the equation x x 0 This is an equation (contains an equal sign). It contains one radical term. Check x x 0 x x ( x ) ( x) x 4x x x 4 0 x( 4 x) 0 x 0 or ( 4x ) 0 x 0 or 4x x 0 or x 4 Check 0 Check 0 0 0 0 0 x : ( ) x : 4 0 4 4 0 4 4 0 0 0 Isolate the radical. Square both sides; remember to square each factor. This is a quadratic equation that can be solved by factoring. Set each factor equal to zero. Solve the two linear equations. There are two possible answers. Substitute each answer into the original equation. Simplify the results to be sure each yields a true statement. Both answers check.
Page 0 of Example: Solve the equation x + x This is an equation (contains an equal sign). It contains one radical term. x + x x + ( x ) x + x 6x + 9 x 7x + 6 0 x x 6 0 ( )( ) x 0 or x 6 0 x or x 6 Check Check x : + 4 Check x 6 : 6+ 6 9 The radical is isolated. Square both sides. Note: ( x ) x. This is a quadratic equation that can be solved by factoring. Set each factor equal to zero. Solve the two linear equations. There are two possible solutions. Substitute each answer into the original equation. Simplify the results to be sure each yields a true statement. Notice x does not yield a true statement, while x 6 yields a true statement. The solution is x 6.
Page of Example: Solve the equation x + x 4 This is an equation (contains an equal sign). It contains two radical terms. Use the steps for equations that contain two radicals. x+ x 4 x ( x )( x ) ( x ) ( x ) ( ) 4 x x x x+ + x 4 x+ + 4 x+ 9+ 6 x 4+ x 4 x 4 6 x 4 ( x ) 6 x 4 ( x ) x 4 x 4 + 4 9( 4) x+ 40 0 5 8 0 ( x 5) 0 or ( x 8) 0 x 5 or x 8 Isolate one radical by adding x 4 to both sides. Square both sides. + x 4 + x 4 Note: ( ) Now there is one radical term left in the equation. Combine like terms and isolate the radical. Factor two out of the terms on the right side. Divide both sides of the equation by a common factor of. Squared both sides. This is a quadratic equation that can be solved by factoring. Set each factor equal to zero. Solve the two linear equations. There are two possible solutions.
Page of Check Check 5 Check 8 x : ( ) x : ( ) 5+ 5 4 5 + 6 4 8+ 8 4 4 + 4 5 4 5 Substitute each answer into the original equation. Simplify the results to be sure each yields a true statement. Both x 5 and x 8 check, so they are both solutions.