Question

What is the value of`x` if`x^(4/5)=(2^8)/(3^8)`?

Answer 1

`x=1024/59049`

Explanation:

If `x > 0` and `a, b` are any real numbers, then:

`(x^a)^b = x^(ab)`

So we find:

`x = x^1 = x^(4/5*5/4) = (x^(4/5))^(5/4)=(2^8/3^8)^(5/4)= ((2/3)^8)^(5/4) = (2/3)^(8*5/4) = (2/3)^10 = 2^10/3^10=1024/59049`

Answer 2

`x=(2/3)^10`

Explanation:

In general if `x^a=p^b` then
`color(white)("XXX")x=(x^a)^(1/a) = (p^b)^(1/a)`

In this case
`color(white)("XXX")a = 4/5color(white)("XX")rarrcolor(white)("XX")1/a=5/4`

`color(white)("XXX")p=2/3`
and
`color(white)("XXXXXX")`after noting that `(2^8)/(3^8)=(2/3)^8`
`color(white)("XXX")b=8`

So
`color(white)("XXX")x=((2/3)^8)^(5/4) = (2/3)^((8 * 5)/4) = (2/3)^10`

Answer 3

`x=0.01734152`

Explanation:

`x^(4/5)=(2^8)/3^8`

Make `x` radical,

`root(5)(x^4)=2^8/3^8`

Multiply both sides by the index of 5,

`(root(5)(x^4))^5=(2^8/3^8)^5`
`x^4=2^40/3^40`

Root both sides by index of 4,

`root(4)(x^4)=root(4)(2^40/3^40)`
`(x^4)^(1/4)=(2^40/3^40)^(1/4)`
`x=2^10/3^10`
`x=1024/59049`

Hence `x=0.01734152`.