How do you evaluate #( \frac { 8} { 27} ) ^ { - \frac { 2} { 3} }# without using a calculator?

Answer 1

#(8/27)^(-2/3)=9/4#

As #x^(-1)=1/x#, #x^(1/n)=root(n)x# and #x^(mn)=(x^m)^n#
#(8/27)^(-2/3)#
= #((8/27)^(-1))^(2/3)#
= #(1/(8/27))^(2/3)#
= #((27/8)^(1/3))^2#
= #(root(3)27/root(3)8)^2#
= #(root(3)(3xx3xx3)/root(3)(2xx2xx2))^2#
= #(3/2)^2#
= #9/4#
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Answer 2

See a solution process below:

First, rewrite this expression as:

#(8/27)^(-2/3) => (8/27)^(1/3 xx -2)#

We can rewrite this using this rule for exponents:

#x^(color(red)(a) xx color(blue)(b)) = (x^color(red)(a))^color(blue)(b)#
#(8/27)^(color(red)(1/3) xx color(blue)(-2)) => ((8/27)^(color(red)(1/3)))^ color(blue)(-2)#

We can now write this in radical form using this rule:

#x^(1/color(red)(n)) = root(color(red)(n))(x)#
#((8/27)^(1/color(red)(3)))^-2 = (root(color(red)(3))(8/27))^-2#

We can now use this rule for dividing radicals to evaluate the radical:

#root(n)(color(red)(a)/color(blue)(b)) = root(n)(color(red)(a))/root(n)(color(blue)(b))#
#(root(3)(color(red)(8)/color(blue)(27)))^-2 => (root(3)(color(red)(8))/root(3)(color(blue)(27)))^-2 => (root(3)(color(red)(2 * 2 * 2))/root(3)(color(blue)(3 * 3 * 3)))^-2 => (2/3)^-2#

Yet again, we can rewrite this as:

#(2/3)^-2 => (2 xx 1/3)^-2 => 2^-2 xx 1/3^-2#

Now, using these rules of exponents, yes, we can rewrite this again and then evaluate:

#x^color(red)(a) = 1/x^color(red)(-a)# and #1/x^color(blue)(a) = x^color(blue)(-a)#
#2^color(red)(-2) xx 1/3^color(blue)(-2) => 1/2^color(red)(- -2) xx 3^color(blue)(- -2) => 1/2^color(red)(2) xx 3^color(blue)(2) => 1/4 xx 9 =>#
#9/4#
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Answer 3

To evaluate ( \left(\frac{8}{27}\right)^{-\frac{2}{3}} ) without using a calculator, follow these steps:

  1. Rewrite ( \frac{8}{27} ) as ( \left(\frac{2}{3}\right)^3 ), since ( 2^3 = 8 ) and ( 3^3 = 27 ).

  2. Raise ( \left(\frac{2}{3}\right)^3 ) to the power of ( -\frac{2}{3} ), which is equivalent to taking the reciprocal and then raising it to the power of ( \frac{2}{3} ).

  3. The reciprocal of ( \left(\frac{2}{3}\right)^3 ) is ( \left(\frac{3}{2}\right)^3 ).

  4. Raise ( \left(\frac{3}{2}\right)^3 ) to the power of ( \frac{2}{3} ).

  5. Cube both the numerator and denominator separately: ( 3^2 = 9 ) and ( 2^2 = 4 ).

  6. The result is ( \left(\frac{3}{2}\right)^2 = \frac{9}{4} ).

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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.

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.

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.

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.

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