Question

Elizabeth claims that the fourth root of 2 can be expressed as 2^m since (2^m)^n = 2

Answer 1

Answer with explanation:

Statement: Elizabeth claims that the fourth root of 2 can be expressed as

`2^m \text {Since} (2^m)^n=2`

Explanation:

This is Possible only when

`m=(1)/(n) \\\\ \text{or}\\\\ n=(1)/(m)\\\\(2^m)^n=(2^m)^{(1)/(m)}=2\\\\\text{or}(2^m)^n=(2^{(1)/(n))^n}\\\\\Rightarrow \text {Fourth root of 2}=2^{(1)/(4)}`

So, 2 can be expressed as in terms of fourth power:

`=(2^4)^{(1)/(4)`

Answer 2

As per the statement:

Elizabeth claims that the fourth root of 2 can be expressed as 2^m

"fourth root of 2" means
`\sqrt[4]{2} = 2^{(1)/(4)}`

then;

`2^{(1)/(4)} = 2^m`

On comparing both sides we get;

`m = (1)/(4)`

Since, it is also given:

`(2^m)^n = 2`

Solve for n;

`(2^{(1)/(4)})^n = 2^{(n)/(4)}`

then;

`2^{(n)/(4)} =2^1`

On comparing both sides we get;

`(n)/(4) = 1`

Multiply 4 both sides we get;

n = 4

Therefore, value of m and n are
`(1)/(4)` and 4