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How do you simplify #(-64)^(-2/3)#?

Answer 1

Hazel Anglin

See a solution process below:

First, rewrite the expression as:

#(-64)^(-2 xx 1/3)#

Next, use this rule of exponents to separate the exponents:

#x^(color(red)(a) xx color(blue)(b)) = (x^color(red)(a))^color(blue)(b)#

#(-64)^(color(red)(-2) xx color(blue)(1/3)) => (-64^color(red)(-2))^color(blue)(1/3)#

Then, use this rule of exponents to eliminate the negative exponent:

#x^color(red)(a) = 1/x^color(red)(-a)#

#(-64^color(red)(-2))^(1/3) => (1/-64^color(red)(- -2))^(1/3) => (1/-64^color(red)(2))^(1/3) =>#

#(1/4096)^(1/3) => 1/16#

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Answer 2

Avery Alexander

To simplify ((-64)^{-\frac{2}{3}}), you first take the reciprocal of the base and then apply the exponent. So, ((-64)^{-\frac{2}{3}} = \frac{1}{(-64)^{\frac{2}{3}}}).

To simplify ((-64)^{\frac{2}{3}}), first find the cube root of the base ((-64)), which is (-4) because ((-4)^3 = -64). Then, raise (-4) to the power of (2) to get (16).

Therefore, ((-64)^{\frac{2}{3}} = 16).

Now, substitute (16) back into the expression (\frac{1}{(-64)^{\frac{2}{3}}}):

(\frac{1}{16}).

So, ((-64)^{-\frac{2}{3}}) simplifies to (\frac{1}{16}).

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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.