Question

For each sentence below, find the value of x that makes each sentence true. 1) (5^1/5)^5 = 25^x x =___ 2) (8^1/3)^2 = 4^x x =___

Answer 1

1)
`x=(1)/(2)`

2)
`x=1`

Explanation:

We are asked to find value of x for each of our given expressions.

1).
`(5^{(1)/(5)})^5=25^x`

Substitute
`25=5^2`:

`(5^{(1)/(5)})^5=(5^2)^x`

Using exponent property
`(a^m)^n=a^(m\cdot n)`, we will get:

`5^{(1)/(5)\cdot 5}=5^(2\cdot x)`

`5^(1)=5^(2x)`

We know when
`a^n=a^(m)`, then
`n=m`. Since base of both exponents is equal, so we can equate them as:

`1=2x`

`2x=1`

`(2x)/(2)=(1)/(2)`

`x=(1)/(2)`

Therefore, the value of x is
`(1)/(2)`.

(2).
`(8^{(1)/(3)})^2=4^x`

Using exponent property
`(a^m)^n=a^(m\cdot n)`, we will get:

`8^{(1)/(3)* 2}=4^x`

`8^{(2)/(3)}=4^x`

Substitute
`8=2^2` and 4 as Substitute
`4=2^2`:

`(2^3)^{(2)/(3)}=(2^2)^x`

Using exponent property
`(a^m)^n=a^(m\cdot n)`, we will get:

`2^{3*(2)/(3)}=2^(2*x)`

`2^(2)=2^(2x)`

We know when
`a^n=a^(m)`, then
`n=m`. Since base of both exponents is equal, so we can equate them as:

`2=2x`

`2x=2`

`(2x)/(2)=(2)/(2)`

`x=1`

Therefore, the value of x is 1.