Name Class Date 14. Simplifying Expressions with Rational Exponents and Radicals Essential Question: How can you write a radical expression as an expression with a rational exponent? Resource Locker Explore Exploring Operations with Rational and Irrational Numbers What happens when you add two rational numbers? Is the result always another rational number or can it be irrational? Will the sum of two irrational numbers always be rational, always be irrational, or can it be either? What about the product of two irrational numbers? These questions are all used to determine whether a set of numbers is closed under an operation. If the sum of two rational numbers is always rational, the set of rational numbers would be said to be closed under addition. The following tables will combine rational and irrational numbers in various ways. The various sums and products should provide a general idea of which sets are closed under the different operations. A Define rational and irrational numbers. B Complete the following addition table. Note that there are both rational and irrational addends. + -π 7 4 0 3-3 -π -π - _ 3 -π 7 4 0 3-3 C Based on the results in the table, will the sum of two rational numbers sometimes, always, or never be a rational number? Module 14 647 Lesson
D What about the sum of two irrational numbers? E And finally, the sum of a rational number and an irrational number? F Now complete the following multiplication table. Similarly, it has both rational and irrational factors. -π 7 4 0 3 3 -π π 7 4 0 _ 3 -π _ 3 _ 3 G H I Based on the results in the table, will the product of two rational numbers sometimes, always, or never be a rational number? What about the product of two irrational numbers? And finally, the product of a rational number and an irrational number? Module 14 648 Lesson
Reflect 1. Prove that the product of two rational numbers is a rational number by confirming the general case.. Discussion Consider the following statement: The product of two rational numbers is an irrational number. Is it a true statement? Justify your answer. Explain 1 Simplifying Multivariable Expressions Containing Radicals As you have seen, to simplify expressions containing radicals, you can rewrite the expressions as powers with rational exponents. You can use properties of exponents. You have already seen the Power of a Power Property of exponents. There are additional properties of exponents that are suggested by the following examples. 3 = ( )( ) = 5 = +3 _ 3 = _ = 1 = 3- ( 3) = ( 3)( 3) = ( )(3 3) = 3 ( _ = _ 3 _ 3 = _ 3 3 = _ 3 3) ( 3 ) = ( ) = ( )( ) = 6 = 3 These relationships are formalized in the table on the following page. Module 14 649 Lesson
Previous lessons have covered the properties of integer exponents. A natural extension of this is to ask if a number can be raised to an exponent that is a rational number. The answer is yes. If we define a n n = a where n is an integer and n 0, we can demonstrate that a m_ n n =( a ) m when m and n are integers and n 0. a m n = a 1 m n.m = (a n ) n = ( a ) m Notice that n a is not defined if n is even and a < 0. Example 1 A Properties of Exponents Let a and b be real numbers and m and n be rational numbers. Product of Powers Propertya m a m = a m+n Quotient of Powers Property a m _ a n = a m-n, a 0 Power of a Product Property (a b) n = a n b n Power of a Quotient Property ( a_ 9 3 (xy) 9 3 (xy) n a b) = _ n Power of a Power Property (a m ) n b n, b 0 = a mn Simplify each expression. Assume all variables are positive. 9_ = (xy) 3 Rewrite using rational exponent. 3 = (xy) Simplify the fraction in the exponent. = x 3 y 3 Power of a Product Property B 5 x x 5 x _ x = x x Rewrite using rational exponents. = x Product of Powers Property Reflect = x Simplify the exponent. = x Rewrite the expression in radical form. 3. Discussion Why is n a not defined when n is even and a < 0? 4. Rewrite the expression -n a so that n has a coefficient of 1. Then state the conditions under which the expression is undefined. Module 14 650 Lesson
Your Turn Simplify each expression. Assume all variables are positive. 5. (x y) 4 4 x y 4 6. _ 8 4 x 6 Explain Simplifying Multivariable Expressions Containing Rational Exponents Use Properties of Rational Exponents to simplify expressions. Example Simplify each expression. Assume all variables are positive. ( A 8 x 9 _ ) 3 ( 8 x 9 ) 3 3 = ( ) 3 ( x 9 _ ) 3 Power of a Product Property = ( 3 _ 3) x (9 _ 3) Power of a Power Property = x 6 Simplify within the parentheses. = 4 x 6 Simplify. B (64 x 1 ) 6 (64 x 1 ) 6 = ( ) ( x 1 ) Power of a Product Property Reflect = ( ) x ( ) Power of a Power Property = x Simplify within the parentheses. = x Simplify. 7. Simplify ( 8 x 9 ) - _ 3. How is it related to the simplified form of ( 8 x 9 ) _ 3 found in example A? Verify the relationship if one exists. Module 14 651 Lesson
Your Turn Simplify each expression. Assume all variables are positive. 8. ( 4 x x 1 ) - 9. ( 4 9 x x 4 ) 1 Explain 3 Simplifying Real-World Expressions with Rational Exponents The relationship between some real-world quantities can be more complicated than a linear or quadratic model can accurately represent. Sometimes, in the most accurate model, the dependent variable is a function of the independent variable raised to a rational exponent. Use the properties of rational exponents to solve the following real-world scenarios. Example 3 Biology Application The approximate number of Calories C that an animal needs each day is given by C = 7 m 3 4, where m is the animal s mass in kilograms. A Find the number of Calories that a 65 kg bear needs each day. To solve this, evaluate the equation when m = 65. C = 7 m 3 4 3 4 = 7 (65) Substitute 65 for m. = 7 ( 4 65 ) 3 = 7 ( 3 4 4 5 ) = 7 (5) 3 = 7 (15) = 9000 Definition of b m n A 65 kilogram bear needs 9000 Calories each day. Module 14 65 Lesson
B A particular panda consumes 1944 Calories each day. How much does this panda weigh? Substitute for C in the original equation and solve for m. C = 7m 3 4 Original equation = 7m 3 4 Substitute for C. _ 4 = m 3 Divide each side by 7. 7 = m 3 4 Simplify. 3 3 = m 3 4 Rewrite the left side as a power. (3 3 ) = (m 3 4 ) Raise both sides to the power. 3 (3 ) = m 3 ( 4 ) Power of a Power Property 3 4 = m Simplify inside the parentheses. m = The panda weighs Simplify. kilograms. Your Turn Solve each real-world scenario. Image Credits: (t) DLILLC/ Corbis; (b) Radius Images/Corbis 10. The speed of light is the product of its frequency f and its wavelength w. In air, the speed of light is 3 10 8 m/s. a. Write an equation for the relationship described above, and then solve this equation for frequency. b. Rewrite this equation with w raised to a negative exponent. c. What is the frequency of violet light when its wavelength is approximately 400 nanometers (1 nm = 10-9 m)? Module 14 653 Lesson
11. Geometry The formula for the surface area of a sphere S in terms of its volume V is S = (4π) _ 3 (3V) 3. What is the surface area of a sphere that has a volume of 36π cm cubed? Leave the answer in terms of π. What do you notice? Elaborate 1. A set of elements is said to be closed under some operation if performance of that operation on elements of the set always produces an element of the set. Examine the set of integers and the set of rational numbers. Is each set closed under each of the following operations: addition, multiplication, division, and subtraction? Provide a counterexample if the set is not closed under an operation. 13. Why are integers closed under multiplication? 14. Is the set of all numbers of the form a x, where a is a positive constant and x is a rational number, closed under multiplication? Justify your answer. 15. Essential Question Check-In How can you write a radical expression with a rational exponent? Module 14 654 Lesson
Evaluate: Homework and Practice 1. Why are the addition and multiplication tables in the Explore activity symmetric about the diagonal from the upper-left corner to the lower-right? For example, why is the entry in the third row of the second column equal to the entry in the second row of the third column? Would a subtraction table be symmetric about the same diagonal? Online Homework Hints and Help Extra Practice. Prove that the rational numbers are closed under addition. Simplify the given expression. 3. 3 (7 x 3 ) 4 4. 3 (8 x 3 ) 5. 3 (8 y 3 ) 4 6 (8 y 3 ) 4 6. 10 0x 7. ( x ) _ y 8. _ 8x _ 3 16x Module 14 655 Lesson
_ 3 9. (0x) 10. 10, 000 4 z + 10, 000 z 11. ( 1 5x x 9 ) - 1. ( 15x 3 ) - _ 3 13. (x) x (x) x x 14. (1,000,000x 6 ) - 3-15. (x y) 3 (x y 4 16. y) 4 _ _ (x _ y) 4 17. (x _ 8 y _ 4 z _ ) 8 18. ( _ z y x ) 8 (x 10 ) _ 5 (x 19. _ 0. -8 ) _ 4 3-1 x x 8 6 x 4 Module 14 656 Lesson
1. (x y) 4 ( _ y _ ). ( y 8 4 ) 3. Biology Biologists use a formula to estimate the mass of a mammal s brain. For a mammal with a mass of m grams, the approximate mass B of the brain, also in grams, is given by B = _ m 3 8. Find the approximate mass of the brain of a mouse that has a mass of 64 grams. 4. Multi-Step Scientists have found that the life span of a mammal living in captivity is related to the mammal s mass. The life span in years L can be approximated by the formula L = 1m _ 5, where m is the mammal s mass in kilograms. How much longer is the life span of a lion compared with that of a wolf? Image Credites: VisionsofAmerica.com/Joe Sohm Typical Mass of Mammals Mammal Mass (kg) Koala 8 Wolf 3 Lion 43 Giraffe 104 Module 14 657 Lesson
Tim and Tom are painters. Use the given information to provide the desired estimate. 5. Tim and Tom use a liters of paint on a large shipping crate. If the next crate they need to paint is similar but has twice the volume, how much paint should they plan on buying? 6. Tim and Tom are painting a crate. Tom paints 10 square feet per minute. They painted a particular crate in 1 day. Tim uses a sprayer and is 4.7 times as fast as Tom. How long would it take them to paint a crate with twice the volume and of similar shape? 7. Determine whether each of the following are rational or irrational. Select the correct answer for each lettered part. a. The product of and 50 Rational Irrational b. The product of _ and _ 5 Rational Irrational c. C = πr evaluated for r = π -1 Rational Irrational d. C = πr evaluated for r = 1 Rational Irrational e. A = πr evaluated for r = π Rational Irrational f. The product of _ π and _ 50π Rational Irrational g. The product of and 9_ Rational Irrational Module 14 658 Lesson
H.O.T. Focus on Higher Order Thinking 8. Explain the Error Jim wrote the following derivation of a term with a rational exponent based on a term consisting of a radicand that is less than 0. ( - n a ) n = a Definition of root ( -a k ) n = a Substituting a k for n a a nk = a 1 Power of a power property nk = 1 Equate exponents. k = n Solve for k. What is his mistake? For what values of n is the proof correct? Correct it. 9. What If? Assume the integers are not closed under addition. a. Are the rational numbers closed under multiplication? b. Are the rational numbers closed under addition? Module 14 659 Lesson
30. Communicate Mathematical Ideas Prove by contradiction that a rational number plus an irrational number is irrational. To do this assume the negation of what you are trying to prove and show how it will logically lead to something contradicting the given. Assume that a rational number plus an irrational number is rational. r 1 + i 1 = r Given r 1 + i 1 - r 1 = r - r 1 Subtract r 1 from both sides. i 1 = r - r 1 Simplify left side. Provide the contradiction statement to finish the proof. 31. Critical Thinking Show that a number raised to the _ power is the same as the cube root of that number. 3 Lesson Performance Task The balls used in soccer, baseball, basketball, and golf are spheres. How much material is needed to make each of the balls in the table? The formula for the surface area of a sphere is S A = 4πr and the formula for the volume of a sphere is V = 4 3 πr 3. Use algebra to find the formula for the surface area of a sphere given its volume. Complete the table with the surface area of each ball. Ball Volume (in cubic inches) Surface Area (in square inches) soccer ball 356.8 baseball 1.8 basketball 455.9 golf ball.48 Module 14 660 Lesson