Solve sqrt[2]{sqrt[3]{64}}=2^n/m

$$\sqrt[ 2 ] { \sqrt[ 3 ] { 64 } } = 2 ^ { n / m }$$

Solve for n

$$n=m$$

$$m\neq 0$$

Steps Using Definition of Logarithm

Calculate \sqrt[3]{64} and get 4.

$$\sqrt[2]{4}=2^{\frac{n}{m}}$$

Calculate \sqrt[2]{4} and get 2.

$$2=2^{\frac{n}{m}}$$

Swap sides so that all variable terms are on the left hand side.

$$2^{\frac{n}{m}}=2$$

Use the rules of exponents and logarithms to solve the equation.

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

Take the logarithm of both sides of the equation.

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

The logarithm of a number raised to a power is the power times the logarithm of the number.

$$\frac{1}{m}n\log(2)=\log(2)$$

Divide both sides by \log(2).

$$\frac{1}{m}n=\frac{\log(2)}{\log(2)}$$

By the change-of-base formula \frac{\log(a)}{\log(b)}=\log_{b}\left(a\right).

$$\frac{1}{m}n=\log_{2}\left(2\right)$$

Divide both sides by m^{-1}.

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

Solve for m

$$m=n$$

$$n\neq 0$$