Evaluate (( square root of 2)/2)^3

${(\frac{\sqrt{2}}{2})}^{3}$

Step 1

Apply the product rule to $\frac{\sqrt{2}}{2}$ .

$\frac{{\sqrt{2}}^{3}}{2^{3}}$

Step 2

Simplify the numerator.

Rewrite ${\sqrt{2}}^{3}$ as ${(2^{3})}^{\frac{1}{2}}$ .

$\frac{\sqrt{2^{3}}}{2^{3}}$

Raise $2$ to the power of $3$ .

$\frac{\sqrt{8}}{2^{3}}$

Rewrite $8$ as $2^{2} \cdot 2$ .

Factor $4$ out of $8$ .

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

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

$\frac{\sqrt{2^{2} \cdot 2}}{2^{3}}$

Pull terms out from under the radical.

$\frac{2 \sqrt{2}}{2^{3}}$

Step 3

Reduce the expression by cancelling the common factors.

Raise $2$ to the power of $3$ .

$\frac{2 \sqrt{2}}{8}$

Cancel the common factor of $2$ and $8$ .

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

$\frac{2 (\sqrt{2})}{8}$

Cancel the common factors.

Factor $2$ out of $8$ .

$\frac{2 \sqrt{2}}{2 \cdot 4}$

Cancel the common factor.

$\frac{2 \sqrt{2}}{2 \cdot 4}$

Rewrite the expression.

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

Step 4

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.35355339 \dots$