8/18/2019 Fractional Powers

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8/18/2019 Fractional Powers

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Fractional powers are useful when we need to calculate roots using a scientic calculator. Forexample to nd 7√ 38 we rewrite this as 381 / 7 which can be evaluated using a scientic calculator.You may need to check your calculator manual to nd the precise way of doing this, probably withthe buttons x y or x 1 /y .

Check that you are using your calculator correctly by conrming that381 / 7 = 1 .6814 (4 dp)

More generally we can write:

a m/n = n√ a m or equivalently n√ a m

Examples

82 / 3 = ( 3√ 8)2 = 2 2 = 4 , and 323 / 5 = ( 5√ 32)3 = 2 3 = 8

Alternatively,

82 / 3 = 3√ 82 = 3√ 64 = 4, and 323 / 5 = 5√ 323 = 5√ 32768 = 8

Exercises

3. Use a calculator to nd, correct to 4 decimal places, a) 5√ 96, b) 4√ 32.

4. Without using a calculator, evaluate a) 43 / 2

, b) 272 / 3

.5. Use the rule

a n

a m = a n − m with n = 0 to prove that a − m =

1a m

.

6. Each of the following expressions can be written as a n . Determine n in each case:

(a) 1

a 5 (b) √ a ×

1a 2

, (c) 1 (d) 1√ a .

Answers

1. (a) x 6 , (b) 1

x12 , (c)

1

t3 , (d) 4

3 , (e) 1

517 .

2. (a) 2− 3 = 123

= 18

, (b) 19

, (c) 16, (d) 32, (e) 64.

3. a) 2.4915, b) 2.3784. 4. a) 43 / 2 = 8 , b) 272 / 3 = 9 .

6. (a) −5 (b) −32 (c) 0 (d) −

12 .

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