Question

How do you simplify `(8/27)^(-2/3)`?

Answer 1

`9/4`

Explanation:

`(8/27)^(-2/3)`
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`=(27/8)^(2/3)`
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The prime factorization of 27 and 8 is:
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`27=9xx3=3xx3xx3=3^3`
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`8=4xx2=2xx2xx2=2^3`

Substituting the factorization on the above fraction we have:
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`(27/8)^(2/3)`
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`=(3^3/2^3)^(2/3)`
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`=((3/2)^3)^(2/3)`
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Applying the property of power of a power that says:
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`color(red)((a^n)^m=a^(mxxn))`
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`((3/2)^3)^(2/3)`
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`=(3/2)^(3xx(2/3))`
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`=(3/2)^2`
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`=9/4`

Answer 2

`=9/4`

Explanation:

There are 3 different processes indicated in this expression.

Laws of indices:

`x^-m = 1/x^m" "and" "(a/b)^-m = (b/a)^m`

The second law is the one we will apply.

Also `x^(p/q) = rootq(x)^p`

`(8/27)^(-2/3) = (27/8)^(+2/3)`

`= root3(27/8)^2" "larr` find the cube roots first

`=(3/2)^2`

`=9/4`