Question

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

Answer 1

Answer:

As per the statement:

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

"fourth root of 2" means $\sqrt[4]{2} = 2^{\frac{1}{4}}$

then;

$2^{\frac{1}{4}} = 2^m$

On comparing both sides we get;

$m = \frac{1}{4}$

Since, it is also given:

$(2^m)^n = 2$

Solve for n;

$(2^{\frac{1}{4}})^n = 2^{\frac{n}{4}}$

then;

$2^{\frac{n}{4}} =2^1$

On comparing both sides we get;

$\frac{n}{4} = 1$

Multiply 4 both sides we get;

n = 4

Therefore, value of m and n are $\frac{1}{4}$ and 4

Answer 2

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

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

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

        $=(2^4)^{\frac{1}{4}$