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4.4 Simplifying Algebraic Expressions Involving Exponents GOAL Simplify algebraic expressions involving powers and radicals. LEARN ABOUT the Math The ratio of the surface area to the volume of microorganisms affects their ability to survive. An organism with a higher surface area-to-volume ratio is more buoyant and uses less of its own energy to remain near the surface of a liquid, where food is more plentiful. Radius (mm) 1 4p 4 a pb 3 1.5 9p 4.5p 2 16p 32 a pb 3 Mike is calculating the surface area-to-volume ratio for different-sized cells. He assumes that the cells are spherical. For a sphere, SA(r) 5 4pr 2 and V(r) 5 43pr 3. He substitutes the value of the radius into each formula and then divides the two expressions to calculate the ratio. 2.5 ? Surface Area/ Volume 25p 125 a pb 6 3 36 p 36 p 3.5 49p 343 a b 6p How can Mike simplify the calculation he uses? Chapter 4 Exponential Functions 231 EXAMPLE Simplify 1 Representing the surface area-to-volume ratio SA(r) , given that SA(r) 5 4pr 2 and V(r) 5 43 pr 3. V(r) Bram’s Solution SA(r) V(r) I used the formulas for SA and V and wrote the ratio. The numerator and denominator have a factor of p, so I divided both by p. I started to simplify the expression by dividing the coefficients. 1 4pr 2 5 4 3 pr 31 5 3r 21 5 3 r a4 4 3 4 5 4 3 5 3b 3 4 The bases of the powers were the same, so I subtracted exponents to simplify the part of the expression involving r. I used a calculator and substituted r 5 2 in the unsimplified ratio first and my simplified expression next. Each version gave me the same answer, so I think that they are equivalent, but the second one took far fewer keystrokes! Reflecting 232 A. How can you use the simplified ratio to explain why the values in Mike’s table kept decreasing? B. Is it necessary to simplify an algebraic expression before you substitute numbers and perform calculations? Explain. C. What are the advantages and disadvantages to simplifying an algebraic expression prior to performing calculations? D. Do the exponent rules used on algebraic expressions work the same way as they do on numerical expressions? Explain by referring to Bram’s work. 4.4 Simplifying Algebraic Expressions Involving Exponents APPLY the Math EXAMPLE Simplify 2 Connecting the exponent rules to the simplification of algebraic expressions (2x23y 2 ) 3 . (x 3 y24 )2 Adnan’s Solution (2 x23y 2 ) 3 (2) 3 (x23 ) 3 ( y 2 ) 3 5 (x 3y24 ) 2 (x 3 ) 2 ( y24 ) 2 5 8x29y 6 x 6y28 5 8x2926y62 (28) 5 8x215y14 EXAMPLE 8y x15 3 I simplified the whole expression by subtracting exponents of terms with the same base. One of the powers had a negative exponent. To write it with positive exponents, I used its reciprocal. 14 5 I used the product-of-powers rule to raise each factor in the numerator to the third power and to square each factor in the denominator. Then I multiplied exponents. Selecting a computational strategy to evaluate an expression Evaluate the expression (x 2n11 ) (x 3n21 ) for x 5 23 and n 5 2. x 2n25 Bonnie’s Solution: Substituting, then Simplifying (x 2n11 ) (x 3n21 ) (23) 2(2) 11 (23) 3(2) 21 5 x 2n25 (23) 2(2) 25 5 5 (23) 5 (23) 5 (23) 21 (2243) (2243) 1 23 I substituted the values for x and n into the expression. Then I evaluated the numerator and denominator separately, before dividing one by the other. 5 2177 147 Chapter 4 Exponential Functions 233 Alana’s Solution: Simplifying, then Substituting Each power had the same base, so I simplified by using exponent rules before I substituted. (x 2n11 ) (x 3n21 ) x 2n25 5 5 x (2n11) 1 (3n21) x 2n25 I added the exponents in the numerator to express it as a single power. x 5n x 2n25 Then I subtracted the exponents in the denominator to divide the powers. 5 x(5n) 2 (2n25) Once I had a single power, I substituted 23 for x and 2 for n and evaluated. 5 x 3n15 5 (23) 3(2) 15 5 (23) 11 5 2177 147 EXAMPLE 4 Simplifying an expression involving powers with rational exponents 1 Simplify (27a 23b 12 ) 3 1 . (16a 28b 12 ) 2 Jane’s Solution 1 (27a23b12 ) 3 1 28 12 2 (16a b ) 1 23 12 3 3 3 5 27 a b 1 28 12 162a 2 b 2 3a21b4 5 24 6 4a b 3 5 a2114b426 4 3 5 a 3b22 4 5 234 3a 3 4b 2 4.4 Simplifying Algebraic Expressions Involving Exponents In the numerator, I applied the 1 exponent 3 to each number or variable inside the parentheses, using the power-of-a-power rule. I did the same in the denominator, 1 applying the exponent 2 to the numbers and variables. I simplified by subtracting the exponents. I expressed the answer with positive exponents. Sometimes it is necessary to express an expression involving radicals using exponents in order to simplify it. EXAMPLE Simplify a 5 Representing an expression involving radicals as a single power 5 8 3 ! x b. !x 3 Albino’s Solution 5 8 3 ! x5 3 x a 3 b 5 a 3b !x x2 8 8 5 (x5 2 16 5 (x10 Since this is a fifth root divided by a square root, I couldn’t write it as a single radical. 3 2 3 2 I changed the radical expressions to exponential form and used exponent rules to simplify. ) 15 10 3 ) 1 10 3 5 (x ) 3 When I got a single power, I converted it to radical form. 5 x10 5 !x 3 10 In Summary Key Idea • Algebraic expressions involving powers containing integer and rational exponents can be simplified with the use of the exponent rules in the same way numerical expressions can be simplified. Need to Know • When evaluating an algebraic expression by substitution, simplify prior to substituting. The answer will be the same if substitution is done prior to simplifying, but the number of calculations will be reduced. • Algebraic expressions involving radicals can often be simplified by changing the expression into exponential form and applying the rules for exponents. CHECK Your Understanding 1. Simplify. Express each answer with positive exponents. a) x 4 (x 3 ) b) ( p23 ) ( p) 5 m5 m23 a24 d) 22 a c) e) ( y 3 ) 2 f) (k 6 ) 22 Chapter 4 Exponential Functions 235 2. Simplify. Express each answer with positive exponents. a) y 10 ( y 4 ) 23 c) (n 24 ) 3 (n 23 ) 24 e) (x21 ) 4x x23 b) (x23 ) 23 (x21 ) 5 d) w 4 (w23 ) (w 22 ) 21 f) (b27 ) 2 b(b25 )b9 x 7 ( y 2) 3 . x 5y 4 a) Substitute x 5 22 and y 5 3 into the expression, and evaluate it. b) Simplify the expression. Then substitute the values for x and y to evaluate it. c) Which method seems more efficient? 3. Consider the expression PRACTISING 4. Simplify. Express answers with positive exponents. a) (pq 2 ) 21 (p 3q 3 ) b) a x 3 22 b y (w 2x) 2 (x 21 ) 2w 3 c) (ab) 22 b5 e) d) m 2n 2 (m 3n 22 ) 2 f) a (ab) 21 22 b a 2b 23 5. Simplify. Express answers with positive exponents. a) (3xy 4 ) 2 (2x 2y) 3 c) (10x) 21y 3 15x 3y23 e) p25 (r 3 ) 2 (p 2r) 2 (p 21 ) 2 b) (2a3 ) 2 4ab2 d) (3m 4n 2 ) 2 12m22n 6 f) a (x 3y) 21 (x 4y 3 ) 21 b (x 2y 23 ) 22 e) a (32x 5 ) 22 0.2 b (x 21 ) 10 6. Simplify. Express answers with positive exponents. K 1 2 a) (x 4 ) 2 (x 6 ) b) 9(c 8 ) 0.5 (16c 12 ) 0.25 1 3 c) d) "25m212 "36m10 (10x 3 ) 2 Ä (10x 6 ) 21 3 "1024x 20 10 f) "512x 27 9 7. Evaluate each expression. Express answers in rational form with positive exponents. 1 a) (16x 6y 4 ) 2 for x 5 2, y 5 1 1 b) (9p22 ) 2 for p 5 3 6p 2 1 c) (81x 4y 6 ) 2 1 8(x 9y 3 ) 3 for x 5 10, y 5 5 1 (25a4 ) 21 2 d) a b for a 5 11, b 5 10 (7a22b) 2 236 4.4 Simplifying Algebraic Expressions Involving Exponents 8. Evaluate. Express answers in rational form with positive exponents. 3 a) ("10 000x) 2 for x 5 16 b) a (4x 3 ) 4 20.5 b for x 5 5 (x 3 ) 6 c) (22a 2b) 23 " 25a 4b 6 for a 5 1, b 5 2 d) (18m 25n 2 ) (32m 2n) for m 5 10, n 5 1 Ä 4mn 23 9. Simplify. Express answers in rational form with positive exponents. !64a12 3 c) a 1.5 26 b (a ) 2 a) (36m4n6 ) 0.5 (81m12n8 ) 0.25 (6x 3 ) 2 (6y 3 ) b b) a (9xy) 6 2 1 3 21 (x 18 ) 6 0.5 b d) a 5 !243x 10 1 (16x 8y 24 ) 4 10. If M 5 , determine values for x and y so that 32x 22 y 8 T a) M 5 1 b) M . 1 c) 0 , M , 1 d) M , 0 11. The volume and surface area of a cylinder are given, respectively, by the A formulas V 5 pr 2h and SA 5 2prh 1 2pr 2. Determine an expression, in simplified form, that represents the surface area-to-volume ratio for a cylinder. b) Calculate the ratio for a radius of 0.8 cm and a height of 12 cm. a) 12. If x 5 22 and y 5 3, write the three expressions in order from least to greatest. y 24 (x 2 ) 23y 23 x 23 ( y 21 ) 22 , , ( y 25 ) (x 5 ) 22 ( y 2 ) (x 23 ) 24 x 25 ( y 24 ) 2 (x 25 ) ( y 4 ) 13. How is simplifying algebraic expressions like simplifying numerical ones? C How is it different? Extending 14. a) The formula for the volume of a sphere of radius r is V(r) 5 43 p r 3. Solve this equation for r. Write two versions, one in radical form and one in exponential form. 3 b) Determine the radius of a sphere with a volume of 256p 3 m . 15. Simplify ! x(x 2n11 ) , x . 0. 3 3n ! x Chapter 4 Exponential Functions 237