Evaluate square root of 45/32

$\sqrt{\frac{45}{32}}$

Step 1

Rewrite $\sqrt{\frac{45}{32}}$ as $\frac{\sqrt{45}}{\sqrt{32}}$ .

$\frac{\sqrt{45}}{\sqrt{32}}$

Step 2

Simplify the numerator.

Rewrite $45$ as $3^{2} \cdot 5$ .

Factor $9$ out of $45$ .

$\frac{\sqrt{9 (5)}}{\sqrt{32}}$

Rewrite $9$ as $3^{2}$ .

$\frac{\sqrt{3^{2} \cdot 5}}{\sqrt{32}}$

Pull terms out from under the radical.

$\frac{3 \sqrt{5}}{\sqrt{32}}$

Step 3

Simplify the denominator.

Rewrite $32$ as $4^{2} \cdot 2$ .

Factor $16$ out of $32$ .

$\frac{3 \sqrt{5}}{\sqrt{16 (2)}}$

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

$\frac{3 \sqrt{5}}{\sqrt{4^{2} \cdot 2}}$

Pull terms out from under the radical.

$\frac{3 \sqrt{5}}{4 \sqrt{2}}$

Step 4

Multiply $\frac{3 \sqrt{5}}{4 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{3 \sqrt{5}}{4 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 5

Combine and simplify the denominator.

Multiply $\frac{3 \sqrt{5}}{4 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{3 \sqrt{5} \sqrt{2}}{4 \sqrt{2} \sqrt{2}}$

Move $\sqrt{2}$ .

$\frac{3 \sqrt{5} \sqrt{2}}{4 (\sqrt{2} \sqrt{2})}$

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

$\frac{3 \sqrt{5} \sqrt{2}}{4 ({\sqrt{2}}^{1} \sqrt{2})}$

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

$\frac{3 \sqrt{5} \sqrt{2}}{4 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}$

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

$\frac{3 \sqrt{5} \sqrt{2}}{4 {\sqrt{2}}^{1 + 1}}$

Add $1$ and $1$ .

$\frac{3 \sqrt{5} \sqrt{2}}{4 {\sqrt{2}}^{2}}$

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

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

$\frac{3 \sqrt{5} \sqrt{2}}{4 {(2^{\frac{1}{2}})}^{2}}$

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

$\frac{3 \sqrt{5} \sqrt{2}}{4 \cdot 2^{\frac{1}{2} \cdot 2}}$

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

$\frac{3 \sqrt{5} \sqrt{2}}{4 \cdot 2^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{3 \sqrt{5} \sqrt{2}}{4 \cdot 2^{\frac{2}{2}}}$

Rewrite the expression.

$\frac{3 \sqrt{5} \sqrt{2}}{4 \cdot 2^{1}}$

Evaluate the exponent.

$\frac{3 \sqrt{5} \sqrt{2}}{4 \cdot 2}$

Step 6

Simplify the numerator.

Combine using the product rule for radicals.

$\frac{3 \sqrt{2 \cdot 5}}{4 \cdot 2}$

Multiply $2$ by $5$ .

$\frac{3 \sqrt{10}}{4 \cdot 2}$

Step 7

Multiply $4$ by $2$ .

$\frac{3 \sqrt{10}}{8}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{3 \sqrt{10}}{8}$

Decimal Form:

$1.18585412 \dots$

Evaluate square root of 45/32

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