How do you simplify this expression? (256/625)^(-3/4)

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(256 / 625) ^-(3/4)

the negative power, -(3/4) can be changed to a positive by placing the expression in the denominator with 1 as the numerator:

1 / ((256 / 625)^(3/4))

Now, the power of 3/4 means that you cube the internal expression and then take the 4th root of that number.  This can be done separately for both the numerator and the denominator of the internal expression:

(256^3)^(1/4) = (256 * 256 * 256)^ 1/4 = 64

(625^3)^(1/4) = (625 * 625 * 625)^1/4 = 125

This leaves the following:

1 / (64 / 125)

and when this expression is carried out:

1 * 125 / 64 = 125 / 64  Answer

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`(256/625)^(-3/4)`

 

Expressions with fractional exponents are written as radicals with an index equal to the denominator of the exponent.

`(root(4)(256))^-3/625^(-3/4)`

Pull out perfect 4th roots from the radical.

`(4)^-3/(625)^(-3/4)`

Remove the negative exponent.

`(1/(4)^3)/625^(-3/4)`

`(1/64)/(625)^(-3/4)`

Now let's work on the denominator. Remember expressions with fractional exponents can be written as a radical with an index equal to the denominator of the exponent.

`(1/64)/(root(4)(625)^-3`

 

Pull out the perfect 4th roots .

`(1/64)/(5)^-3`

 

Remove the negative exponent.

`(1/64)/(1/(5)^3`

 `(1/64)/(1/125)`

 `125* (1/64)`

 

125/64

 

 

 

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