SIMPLIFYING SQUARE ROOTS


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1 40 (88) Chapter 8 Powers and Roots 8. SIMPLIFYING SQUARE ROOTS In this section Using the Product Rule Rationalizing the Denominator Simplified Form of a Square Root In Section 8. you learned to simplify some radical expressions using the product rule. In this section you will learn three basic rules to follow for writing expressions involving square roots in simplest form. These rules can be extended to radicals with index greater than, but we will not do that in this text. Using the Product Rule We can use the product rule to simplify square roots of certain numbers. For example, 4 9 Factor 4 as Because 4 is not a perfect square, we cannot write 4 without the radical symbol. However, is considered a simpler expression that represents the exact value of 4. When simplifying square roots, we can factor the perfect squares out of the radical and replace them with their square roots. Look for the factors 4, 9,,,, 49, and so on. E X A M P L E calculator closeup You can use a calculator to see that and agree for the first 0 digits (out of infinitely many). Having the same first 0 digits does not make =. The product rule for radicals guarantees that they are equal. Simplifying radicals using the product rule Simplify. a) b) 0 c) a) Because 4, we can use the product rule to write 4. b) 0 c) Note that 4, 9, and are perfect squares and are factors of. In factoring out a perfect square, it is most efficient to use the largest perfect square: If we had factored out 9, we could still get the correct answer as follows: Rationalizing the Denominator Radicals such as,, and are irrational numbers. So a fraction such as has an irrational denominator. Because fractions with rational denominators are considered simpler than fractions with irrational denominators, we usually convert fractions with irrational denominators to equivalent ones with rational denominators. That is, we rationalize the denominator.
2 8. Simplifying Square Roots (89) 4 E X A M P L E Rationalizing denominators Simplify each expression by rationalizing its denominator. a) b) a) Because, we multiply numerator and denominator by : Multiply numerator and denominator by. b) Because, multiply the numerator and denominator by : Multiply numerator and denominator by. Simplified Form of a Square Root When we simplify any expression, we try to write a simpler expression that is equivalent to the original. However, one person s idea of simpler is sometimes different from another person s. For a square root the expression must satisfy three conditions to be in simplified form. These three conditions provide specific rules to follow for simplifying square roots. Simplified Form for Square Roots An expression involving a square root is in simplified form if it has. no perfectsquare factors inside the radical,. no fractions inside the radical, and. no radicals in the denominator. Because a decimal is a form of a fraction, a simplified square root should not contain any decimal numbers. Also, a simplified expression should use the fewest number of radicals possible. So we write rather than even though both and are both in simplified form. E X A M P L E Simplified form for square roots Write each radical expression in simplified form. a) 00 b) c) 0 a) We must remove the perfect square factor of 00 from inside the radical:
3 4 (80) Chapter 8 Powers and Roots calculator b) We first use the quotient rule to remove the fraction from inside the radical: closeup Using a calculator to check simplification problems will help you to understand the concepts. 0 Quotient rule for radicals c) The numerator and denominator have a common factor of : 0 Reduce. 0 Note that we could have simplified by first using the quotient rule to get 0 0 and then reducing 0. Another way to simplify 0 is to first multiply the numerator and denominator by. You should try these alternatives. Of course, the simplified form is by any method. In the next example we simplify some expressions involving variables. Remember that any exponential expression with an even exponent is a perfect square. E X A M P L E 4 Radicals containing variables Simplify each expression. All variables represent nonnegative real numbers. a) x b) 8a 9 c) 8a 4 b a) x x x The largest perfect square factor of x is x. x x xx For any nonnegative x, x x. b) 8a 9 4a 8 a The largest perfect square factor of 8a 9 is 4a 8. a 4 a 4a 8 a 4 c) 8a 4 b 9a 4 b b Factor out the perfect squares. a b b 9a 4 b a b
4 8. Simplifying Square Roots (8) 4 If square roots of variables appear in the denominator, then we rationalize the denominator. E X A M P L E helpful hint If you are going to compute the value of a radical expression with a calculator, it doesn t matter if the denominator is rational. However, rationalizing the denominator provides another opportunity to practice building up the denominator of a fraction and multiplying radicals. Radicals containing variables Simplify each expression. All variables represent positive real numbers. a) a b) a b c) a a a) a a a a a b) a b a b a b b b ab b Multiply numerator and denominator by a. a a a Quotient rule for radicals a c) a a a a a 4 a a a 4 a Factor out the perfect square. a a a a Factor the denominator. Divide out the common factor. CAUTION Do not attempt to reduce an expression like the one in Example (c): a a You cannot divide out common factors when one is inside a radical.
5 44 (8) Chapter 8 Powers and Roots WARMUPS True or false? Explain your answer.. 0 True. 8 9 False. True False. a aa for any positive value of a. True. a 9 a for any positive value of a. False. y y 8 y for any positive value of y. True 8. False 9. 4 False 0. 8 False 8. EXERCISES Reading and Writing After reading this section, write out the answers to these questions. Use complete sentences.. How do we simplify a radical with the product rule? We use the product rule to factor out a perfect square from inside a square root.. Which integers are perfect squares? The perfect squares are, 4, 9,,, and so on.. What does it mean to rationalize a denominator? To rationalize a denominator means to rewrite the expression so that the denominator is a rational number. 4. What is simplified form for a square root? A square root in simplified form has no perfect squares or fractions inside the radical and no radicals in the denominator.. How do you simplify a square root that contains a variable? To simplify a square root containing variables, use the same techniques as we use on square roots of numbers.. How can you tell if an exponential expression is a perfect square? Any even power of a variable is a perfect square. Assume that all variables in the exercises represent positive real numbers. Simplify each radical. See Example Simplify each expression by rationalizing the denominator. See Example Write each radical expression in simplified form. See Example
6 8. Simplifying Square Roots (8) 4 Simplify each expression. See Example a y 0 4. a 9 a 4 y a 4 a 4. t 4. 8a 48. 8w 9 t t a w 4 w 49. 0a 4 b 9 0. xy. xy a b 4 b xyy xyxy. 4xy. a b 8 c 4. xy z 9 4 x yxy ab 4 ca xy 4 z xy Simplify each expression. See Example.... x x a x x a x x a b y 0x 0b b y y x x. x y. w. 0 y x xy 0w xy y w x 4. 4 x y. 8 y xy y x xxy y s. 8 t s st t Simplify each expression.. 80x 8. 90y yx 9 4xx y 40 0 y 4 x yx 0x x y 0. 48xy.. x x 4xy y 4x x x yx p. 4. 0t. a b a b c 4 p p q t t a b c pq q 0t t a b 8 cac. n4 b n b c 4xy. 8m 8. n nb c x 9 y xy m n mn n 4 b c nb y n x 9 4 y m n Solve each problem. 8. Economic order quantity. The formula for economic order quantity E A I S was used in Exercise 8 of Section 8.. a) Express the righthand side in simplified form. E AIS I b) Find E when A, S $4, and I $80.. FIGURE FOR EXERCISE Landing speed. Aircraft design engineers determine the proper landing speed V (in ft/sec) by using the formula V 84L, CS where L is the gross weight of the aircraft in pounds, C is the coefficient of lift, and S is the wing surface area in square feet. a) Express the righthand side in simplified form. V 9 LCS CS b) Find V when L 800 pounds, C.8, and S 00 square feet..4 Use a calculator to evaluate each expression FIGURE FOR EXERCISE 84
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