6. Radical and Rational Exponent of a number? How can you write and evaluate an nth root Recall that you cube a number a follow. Symbol for cubing i rd power. = = 8 cubed i 8. To undo thi, take the cube root of the number. Symbol for cube root i. 8 = = The cube root of 8 i. ACTIVITY: Finding Cube Root Work with a partner. Ue a cube root ymbol to write the ide length of the cube. Then find the cube root. Check your anwer by multiplying. Which cube i the larget? Which two are the ame ize? Explain your reaoning. a. Volume 7 ft b. Volume 5 cm c. Volume 75 in. COMMON CORE Exponent In thi leon, you will implify expreion with rational exponent. Learning Standard N.RN. N.RN. d. Volume.75 m e. Volume yd 5 f. Volume mm 8 Cube are not drawn to cale. 76 Chapter 6 Exponential Equation and Function
ACTIVITY: Etimating nth Root Work with a partner. When you raie an nth root of a number to the nth power, you get the original number. ( n a ) n = a Sample: The th root of 6 i becaue = 6. 6 = Check: = = 6 Match the nth root with the point on the number line. Jutify your anwer. a. 5 b. 0.5 c. 5.5 d. 65 e. 55 f. 6 0,000 A. B. C. D. E. F. 0 5 6 Math Practice Jutify Concluion What information can you ue to jutify your anwer?. IN YOUR OWN WORDS How can you write and evaluate the nth root of a number?. The body ma m (in kilogram) of a dinoaur that walked on two feet can be modeled by m = (0.0006)C.7 where C i the circumference (in millimeter) of the dinoaur femur. The ma of a Femur Tyrannoauru rex wa 000 kilogram. What wa the circumference of it femur? Ue what you learned about cube root to complete Exercie 5 on page 80. Section 6. Radical and Rational Exponent 77
6. Leon Leon Tutorial Key Vocabulary nth root, p. 78 When b n = a for an integer n greater than, b i an nth root of a. n a nth root of a The nth root of a number may be real number or imaginary number. You will tudy imaginary number in a future coure. Study Tip In Example b, although = 8 and ( ) = 8, 8 = becaue the radical ymbol indicate the poitive root. Finding nth Root Simplify each expreion. a. 6 b. 6 = 8 = = 8 = Rational Exponent Word The nth root of a poitive number a can be written a a power with bae a and an exponent of /n. Number 8 = 8 / Algebra n a = a /n Reading When n =, the i typically not written with the radical ign. Simplifying Expreion with Rational Exponent Simplify each expreion. a. 00 / 00 / = 00 Write the expreion in radical form. = 0 0 Rewrite. = 0 Simplify. b. /5 /5 = 5 Write the expreion in radical form. = 5 Rewrite. = Simplify. Exercie 8 Simplify the expreion.. 6. 5. 65. 9 / 5. / 6. 6 /6 78 Chapter 6 Exponential Equation and Function
You can ue propertie of exponent to implify expreion involving rational exponent. Uing Propertie of Exponent a. 6 / = 6 (/) Rewrite the exponent. = (6 / ) Power of a Power Property = Evaluate the fourth root of 6. = 8 Evaluate power. b. 7 / = 7 / Rewrite the exponent. = (7 / ) Power of a Power Property = Evaluate the third root of 7. = 8 Evaluate power. Exercie 0 5 Simplify the expreion. 7. 6 / 8. 9 5/ 9. 56 / Real-Life Application Volume 5 cubic feet The radiu r of a phere i given by the equation r = ( V π ) /, where V i the volume of the phere. Find the radiu of the beach ball to the nearet foot. Ue. for π. r = ( V π) / = [ () (.) ] / = (.56) 9 / Write the equation. Subtitute for V and. for π. Multiply. Ue a calculator. The radiu of the beach ball i about feet. 0. WHAT IF? In Example, the volume of the beach ball i 7,000 cubic inche. Find the radiu to the nearet inch. Ue. for π. Section 6. Radical and Rational Exponent 79
6. Exercie Help with Homework. WRITING Explain how to implify 8 /.. WHICH ONE DOESN T BELONG? Which expreion doe not belong with the other three? Explain your reaoning. ( 7 ) 7 / 7 / 9+(-6)= +(-)= +(-9)= 9+(-)= Find the dimenion of the cube. Check your anwer.. Volume 6 in.. Volume 6 cm 5. Volume ft 5 Write the expreion in rational exponent form. 6. 7 5 7. ( ) 8. ( 5 8 ) Write the expreion in radical form. 9. 5 / 0. 0 /7. 78 /5. ERROR ANALYSIS Decribe and correct the error in writing the expreion in rational exponent form. ( ) = / Simplify the expreion.. 56. 5 5. 5 0 6. 8 /7 7. 000 / 8. 8 / 9. BAKE SALE A math club i having a bake ale. Find the length and width of the bake ale ign. / ft Math Club Bake Sale thi Saturday 6 79 ft 80 Chapter 6 Exponential Equation and Function
Simplify the expreion. 0. /5. 5 /. 6 /. /5. 8 5/7 5. / 6. PAPER CUPS The radiu r of the bae of a cone i given by the equation r = ( V πh) /, where V i the volume of the cone and h i the height of the cone. Find the radiu of the paper cup to the nearet inch. Ue. for π. 7. WRITING Explain how to write ( n a ) m in rational exponent form. in. Volume 5 in. 8. PROBLEM SOLVING The formula for the volume of a regular dodecahedron i V 7.66, where i the length of an edge. The volume of the dodecahedron i 0 cubic feet. Etimate the edge length. Logic Determine whether the tatement i alway, ometime, or never true. Let x be a nonnegative real number. Jutify your anwer. 9. ( x / ) = x 0. x / = x. x / = x. x / = x. x / x / = x. x = x / x Graph the linear equation. (Section. and Section.) 5. y = x + 6. x y = 6 7. y = x 5 8. MULTIPLE CHOICE Which equation i hown in the graph? (Section.) A y = x + B y = x C y = x D y = x + y x Section 6. Radical and Rational Exponent 8