# 2.2. Multiplying and Dividing Powers. INVESTIGATE the Math

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1 . Multipling and Dividing Powers GOAL Develop and appl exponent principles to multipl and divide powers. INVESTIGATE the Math Amir thought there was a wa to simplif without using a calculator.? How could Amir simplif this expression without a calculator? A. Draw a table like the one shown. Use it to record the products of powers that have the same base but different exponents. Extend the table until ou see a pattern. Use the pattern to determine how to quickl multipl powers with the same base. Product Expressed Multiplication Expanded Form as a Single Power B. How is the exponent of the product related to the exponents of the factors? C. Test our answer to part B using these expressions. Check using a calculator. a) b) D. Draw a new table like the one shown. Use it to record quotients with the same base and different positive integers for exponents. Continue to add rows to the table until ou see a pattern. Use the pattern to determine how to quickl divide powers with the same base. Quotient Simplified and Expressed Division Expanded Form as a Single Power E. How is the exponent of the quotient related to the exponents of the terms in the division statement? NEL Powers and Polnomials 8

2 F. Use our results in parts B and E to simplif as a single power 1 with base. Check our answer b calculating the value of the original and simplified expressions. Reflecting G. How can ou determine the exponent of the product of powers with the same base? H. How can ou determine the exponent of the quotient of powers with the same base? principle a basic truth or rule about the wa something works I. Wh do the principles in parts G and H onl appl to products and quotients of powers with the same base? APPLY the Math EXAMPLE 1 Representing an expression involving powers 1x 7 1x Simplif. 6 x Ton's Solution: Making a conjecture based on numeric examples I substituted for x and calculated the value of the expression. 16 is, which equals 4. Then, I substituted for x and calculated the value of the expression. 81 is, which equals Chapter NEL

3 . I think that 1x 7 1x x 4 6 x In both cases, the final power had an exponent of 4. I thought the same pattern would work for an base. Dann s Solution: Reasoning using the definition of a power 1x 7 1x 1xxxxxxx1xxx 1 1xxxxxx1xxxx xxxxxx 1 xxxx x 4 I looked at the numerator and wrote out the powers in expanded form. This showed me that the numerator was x 10. I wrote out the denominator in expanded form. I simplified the fraction b dividing 6 of the xs in the numerator and denominator. This left 4 xs multiplied together, which I wrote as x 4. 1x 7 1x So, x 4. Patt s Solution: Reasoning using exponent principles a x 7 x b x 6 1x1x 1x1x 4 x 1x 7 1x x 76 x 4 I divided x 7 b using the exponent principle for division. Subtracting 6 from 7, I got 1. I didn t need to write the exponent because x 1 is the same as x. I used the exponent principle for multiplication. Since 1 is 4, m final answer was x 4. I saw that I could have reached the same answer in one step b just adding the exponents in the numerator and subtracting the exponent in the denominator. Communication Tip An variable without a visible exponent is understood to have an exponent of 1. For example, x x 1, 1, and 4c 4c 1. NEL Powers and Polnomials 85

4 EXAMPLE Selecting a strateg to evaluate an expression with two variables 1x 4 1x 5 Simplif and evaluate when x and. x 5 5 Kathrn s Solution: Using an algebraic strateg to first simplif the expression 1x 4 1x 5 x 5 5 1x4 x 1 5 x 5 5 x 7 8 x 5 5 x x Everthing in the numerator was multiplied together. I rewrote it so that powers of the same base were side b side. Then, I simplified b adding the exponents of the powers having the same base. I divided powers having the same base. I did this b subtracting the exponent of the denominator from the exponent of the numerator. I substituted for x and for, and evaluated. The result was 7. Jerem s Solution: Using a substitution strateg 1x 4 1x 5 x I substituted for x and for into the question. I calculated the expressions inside each set of brackets. I did the final multiplication and division. 86 Chapter NEL

5 . EXAMPLE Appling exponent principles to simplif expressions where the base has multiple factors 5 5 x b Simplif. 5 b x Miranda's Solution: Connecting exponent principles to fraction operations 5 5 x b 5 b x 5 5 b x b x 15 5 b x b x x 4 4 I expressed the numerator as three separate powers because I knew that the exponent 5 outside of the brackets referred to each of the factors inside. Then, I divided powers having the same base b subtracting their exponents. I knew that x and in the denominator each had an exponent of 1. I evaluated b 5 b multipling.. 5 ba 5 b 4 5 Communication Tip You can rewrite a product or quotient raised to an exponent b appling the exponent to each of the terms. For example, and a x 1x x. b x If each rational number in an algebraic expression can be expressed as a terminating decimal, ou can write each number as such and then simplif the equivalent expression. NEL Powers and Polnomials 87

6 Lee s Solution: Connecting exponent principles to decimal operations 10.4 x x x x x x x 4 4 I converted each fraction to its decimal equivalent. Then, I expressed the numerator as three separate powers because I knew that the exponent 5 outside the brackets referred to each of the factors inside. I divided powers having the same base b subtracting their exponents. The x and in the denominator each had an exponent of 1. I evaluated The final result was 0.16 x 4 4. EXAMPLE 4 Using exponent principles to solve a problem involving large numbers The M1 galax in the constellation of Andromeda is about km awa. Light travels at about km/ear. Estimate how long it would take light to reach Earth from M1. Kle s Solution I used the formula t d. s d km s km>ear t I estimated this to be about I needed to divide the distance to the M1 galax b the speed of light to find the time in ears is close to 10 19, is close to , and is I used the exponent principle 10 1 for quotients to simplif the expression Chapter NEL

7 , so The light from M1 takes about ears to reach Earth! 10 6 In Summar Ke Idea When two powers have the same base, these principles can be used to simplif their product or quotient: Exponent Principle for Products 1a m 1a n a mn Exponent Principle for Quotients 1a m 1a n a mn 1 0 For example, and 4 4. Need to Know These exponent principles onl work when the powers involved have the same base. It is more efficient to simplif an algebraic expression involving powers before substituting to evaluate it. An exponent applied to a product or quotient can be written b appling the exponent to each of its terms. In general 1ab m a m b m and a a m. b b am m 1b 0 b CHECK Your Understanding 1. Simplif. a) 1 1 b). Simplif. 5 a) b). Simplif if possible, and then evaluate a) b) Simplif, and then evaluate for x and 5. 1x 4 1x 6 x 4 a) b) x 1x 4 1x 6 NEL Powers and Polnomials 89

8 PRACTISING 5. Simplif. a) c) b) 1m 4 1m d) x 4 1x a 5 b a 5 b a 5 b 4 6. Simplif. a) 1n 5 1w 6 1n 1w 7 d) 1b 5 4 1b 1b 7 b) 1m 1m 4 1r 8 1m 1r e) 1x 4 11x 1 c) 5 1 p 1 p 1 p 8 f ) 1a 5 1 1a 4 1a1 7. Wh do ou get the same result for each of these expressions? a) c) b) d) Simplif. a) m x b) c) d) 5 m 1 4 1x Simplif. a) x x a 17 1x 5 b) c) d) 17 a x x 4 x Create four different expressions involving exponents that simplif to Simplif if possible, and then evaluate. K a) c) e) a 7 b 4 a 7 b b) d) f ) Simplif, and then evaluate for x and 5. 1x 5 1x x 4 a) c) e) x 8 1x b) d) f ) a 4 5 b 5 a 4 5 b 4 a 4 5 b 6 61x x a 4 x b a 4 b x 90 Chapter NEL

9 1. If ou know that the product of two powers is 7 10 and that the A quotient is 7, what could the two powers be? How could ou verif our answer? 14. Scientists estimate that there are cells in the average human. T There are approximatel humans in the world. Approximatel how man cells do all the humans on Earth have? Write our answer using a power with base Explain wh it is necessar for the bases to be the same in order to C appl the multiplication and division principles for exponents.. Extending 16. a) Complete the table to show the relationship between the metric units of length. Express each relationship as a power with base 10. Millimetres Centimetres Metres Kilometres Millimetres Centimetres Metres Kilometres b) Determine the number of centimetres in 5 km. c) Determine the number of millimetres in 4 m. 17. A piece of steel plate is used to make a railwa car. The plate is.5 m wide,. m long, and 0.5 cm thick. Determine the volume of steel in cubic centimetres. 18. The annual worldwide production of all grains is about kg. How much grain is produced per person if there are approximatel people in the world? a) Evaluate. b) Simplif using the exponent principle for quotients. c) Use the meaning of the powers in to simplif the expression. d) Discuss what 0 might mean. e) Discuss whether a 0 would have a similar meaning for an value of a. 5 5 NEL Powers and Polnomials 91

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