Question

How do you simplify `(1/8) ^ (5/2)`?

Answer 1

`(1/8)^(5/2)=(1/2)^(15/2)=sqrt2/256`

Explanation:

We can first work this and keep it in terms of exponents. Since `1=1^3 and 8=2^3`, we can write:

`(1/8)^(5/2)=(1^3/2^3)^(5/2)=((1/2)^3)^(5/2)=(1/2)^(3xx5/2)=(1/2)^(15/2)`

(this also uses the rule that `(x^a)^b=x^(ab)`)

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We can also evaluate this numerically.

Let's first see that the exponent `15/2` means that we're going to take the fraction `1/2` and take the square root (that's the 2 in `15/2`) and also take `1/2` to the 15th power (that's the 15 in `15/2`).

Let's talk about the numerator first.

`1^(15/2)=1`

And now the denominator:

`(2)^(15/2)=2^(14/2+1/2)=2^(7+1/2)=2^7xx2^(1/2)=128sqrt2`

(we used the rule that `x^a xx x^b=x^(a+b)`)

We can then say that:

`(1/2)^(15/2)=1/(128sqrt2)`

We can then rationalize the denominator:

`1/(128sqrt2)(sqrt2/sqrt2)=sqrt2/(128xx2)=sqrt2/256`