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How do you simplify #4(8)^(-2/3)#?

Answer 1

Kennedy Adams

The negative index means that the base can be moved to the denominator and become positive.

#4(8)^(-2/3)#

#= 4/8^(2/3)#

A fractional index can be written as an index and a root.

#=4/(root3(8)^2)#

#=4/2^2#

#=4/4#

#=1#

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Answer 2

Eva Abramson

To simplify (4(8)^{-\frac{2}{3}}), we first rewrite (8) as (2^3), since (8) is (2) raised to the power of (3). Then we apply the properties of exponents:

[4(8)^{-\frac{2}{3}} = 4(2^3)^{-\frac{2}{3}}]

Next, we apply the power of a power property, which states that ((a^m)^n = a^{m \times n}):

[4(2^3)^{-\frac{2}{3}} = 4 \times 2^{3 \times -\frac{2}{3}}]

Now, we multiply the exponents:

[3 \times -\frac{2}{3} = -2]

So we have:

[4 \times 2^{-2}]

Finally, we simplify (2^{-2}) to (\frac{1}{2^2}) or (\frac{1}{4}):

[4 \times \frac{1}{4} = 1]

Therefore, (4(8)^{-\frac{2}{3}}) simplifies to (1).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.