Simplify ((4^(1/2))/(4^(3/4)))^4

${(\frac{4^{\frac{1}{2}}}{4^{\frac{3}{4}}})}^{4}$

Move $4^{\frac{1}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

${(\frac{1}{4^{\frac{3}{4}} \cdot 4^{- \frac{1}{2}}})}^{4}$

Multiply $4^{\frac{3}{4}}$ by $4^{- \frac{1}{2}}$ by adding the exponents.

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

${(\frac{1}{4^{\frac{3}{4} - \frac{1}{2}}})}^{4}$

To write $- \frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

${(\frac{1}{4^{\frac{3}{4} - \frac{1}{2} \cdot \frac{2}{2}}})}^{4}$

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{1}{2}$ by $\frac{2}{2}$ .

${(\frac{1}{4^{\frac{3}{4} - \frac{2}{2 \cdot 2}}})}^{4}$

Multiply $2$ by $2$ .

${(\frac{1}{4^{\frac{3}{4} - \frac{2}{4}}})}^{4}$

Combine the numerators over the common denominator.

${(\frac{1}{4^{\frac{3 - 2}{4}}})}^{4}$

Subtract $2$ from $3$ .

${(\frac{1}{4^{\frac{1}{4}}})}^{4}$

Simplify the expression.

Apply the product rule to $\frac{1}{4^{\frac{1}{4}}}$ .

$\frac{1^{4}}{{(4^{\frac{1}{4}})}^{4}}$

One to any power is one.

$\frac{1}{{(4^{\frac{1}{4}})}^{4}}$

Simplify the denominator.

Multiply the exponents in ${(4^{\frac{1}{4}})}^{4}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1}{4^{\frac{1}{4} \cdot 4}}$

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{1}{4^{\frac{1}{4} \cdot 4}}$

Rewrite the expression.

$\frac{1}{4^{1}}$

Evaluate the exponent.

$\frac{1}{4}$

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{4}$

Decimal Form:

$0.25$