Algebra Examples

${(\frac{4^{\frac{5}{4}} \cdot 4^{\frac{1}{4}}}{4^{\frac{1}{2}}})}^{\frac{1}{2}}$

Step 1

Multiply $4^{\frac{5}{4}}$ by $4^{\frac{1}{4}}$ by adding the exponents.

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Step 1.1

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

${(\frac{4^{\frac{5}{4} + \frac{1}{4}}}{4^{\frac{1}{2}}})}^{\frac{1}{2}}$

Step 1.2

Combine the numerators over the common denominator.

${(\frac{4^{\frac{5 + 1}{4}}}{4^{\frac{1}{2}}})}^{\frac{1}{2}}$

Step 1.3

Add $5$ and $1$ .

${(\frac{4^{\frac{6}{4}}}{4^{\frac{1}{2}}})}^{\frac{1}{2}}$

Step 1.4

Cancel the common factor of $6$ and $4$ .

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Step 1.4.1

Factor $2$ out of $6$ .

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

Step 1.4.2

Cancel the common factors.

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Step 1.4.2.1

Factor $2$ out of $4$ .

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

Step 1.4.2.2

Cancel the common factor.

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

Step 1.4.2.3

Rewrite the expression.

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

Step 2

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

Combine the numerators over the common denominator.

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

Step 3.3

Subtract $1$ from $3$ .

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

Step 3.4

Divide $2$ by $2$ .

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

Step 4

Simplify $4^{1}$ .

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

Step 5

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$\sqrt{4^{1}}$

Step 6

Anything raised to $1$ is the base itself.

$\sqrt{4}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\sqrt{4}$

Decimal Form:

$2$