Algebra Examples

$\sqrt{\frac{32}{\sqrt{2}}}$

Step 1

Multiply $\frac{32}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\sqrt{\frac{32}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}}$

Step 2

Combine and simplify the denominator.

Tap for more steps...

Step 2.1

Multiply $\frac{32}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\sqrt{\frac{32 \sqrt{2}}{\sqrt{2} \sqrt{2}}}$

Step 2.2

Raise $\sqrt{2}$ to the power of $1$ .

$\sqrt{\frac{32 \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}}$

Step 2.3

Raise $\sqrt{2}$ to the power of $1$ .

$\sqrt{\frac{32 \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}}$

Step 2.4

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

$\sqrt{\frac{32 \sqrt{2}}{{\sqrt{2}}^{1 + 1}}}$

Step 2.5

Add $1$ and $1$ .

$\sqrt{\frac{32 \sqrt{2}}{{\sqrt{2}}^{2}}}$

Step 2.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Tap for more steps...

Step 2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\sqrt{\frac{32 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}}$

Step 2.6.2

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

$\sqrt{\frac{32 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}}$

Step 2.6.3

Combine $\frac{1}{2}$ and $2$ .

$\sqrt{\frac{32 \sqrt{2}}{2^{\frac{2}{2}}}}$

Step 2.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.6.4.1

Cancel the common factor.

$\sqrt{\frac{32 \sqrt{2}}{2^{\frac{2}{2}}}}$

Step 2.6.4.2

Rewrite the expression.

$\sqrt{\frac{32 \sqrt{2}}{2^{1}}}$

Step 2.6.5

Evaluate the exponent.

$\sqrt{\frac{32 \sqrt{2}}{2}}$

Step 3

Cancel the common factor of $32$ and $2$ .

Tap for more steps...

Step 3.1

Factor $2$ out of $32 \sqrt{2}$ .

$\sqrt{\frac{2 (16 \sqrt{2})}{2}}$

Step 3.2

Cancel the common factors.

Tap for more steps...

Step 3.2.1

Factor $2$ out of $2$ .

$\sqrt{\frac{2 (16 \sqrt{2})}{2 (1)}}$

Step 3.2.2

Cancel the common factor.

$\sqrt{\frac{2 (16 \sqrt{2})}{2 \cdot 1}}$

Step 3.2.3

Rewrite the expression.

$\sqrt{\frac{16 \sqrt{2}}{1}}$

Step 3.2.4

Divide $16 \sqrt{2}$ by $1$ .

$\sqrt{16 \sqrt{2}}$

Step 4

Rewrite $16 \sqrt{2}$ as ${(2^{2})}^{2} \sqrt{2}$ .

Tap for more steps...

Step 4.1

Rewrite $16$ as $4^{2}$ .

$\sqrt{4^{2} \sqrt{2}}$

Step 4.2

Rewrite $4$ as $2^{2}$ .

$\sqrt{{(2^{2})}^{2} \sqrt{2}}$

Step 5

Pull terms out from under the radical.

$2^{2} \sqrt{\sqrt{2}}$

Step 6

Raise $2$ to the power of $2$ .

$4 \sqrt{\sqrt{2}}$

Step 7

Rewrite $\sqrt{\sqrt{2}}$ as $\sqrt[4]{2}$ .

$4 \sqrt[4]{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$4 \sqrt[4]{2}$

Decimal Form:

$4.75682846 \dots$