Basic Math Examples

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

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

Step 1

Rewrite

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

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

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

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

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

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

Step 2

Simplify the numerator.

Tap for more steps...

Step 2.1

Rewrite

45

$45$ as

3^{2} \cdot 5

$3^{2} \cdot 5$ .

Tap for more steps...

Step 2.1.1

Factor

9

$9$ out of

45

$45$ .

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

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

Step 2.1.2

Rewrite

9

$9$ as

3^{2}

$3^{2}$ .

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

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

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

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

Step 2.2

Pull terms out from under the radical.

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

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

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

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

Step 3

Simplify the denominator.

Tap for more steps...

Step 3.1

Rewrite

32

$32$ as

4^{2} \cdot 2

$4^{2} \cdot 2$ .

Tap for more steps...

Step 3.1.1

Factor

16

$16$ out of

32

$32$ .

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

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

Step 3.1.2

Rewrite

16

$16$ as

4^{2}

$4^{2}$ .

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

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

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

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

Step 3.2

Pull terms out from under the radical.

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

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

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

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

Step 4

Multiply

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

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

\frac{\sqrt{2}}{\sqrt{2}}

$\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Tap for more steps...

Step 5.1

Multiply

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

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

\frac{\sqrt{2}}{\sqrt{2}}

$\frac{\sqrt{2}}{\sqrt{2}}$ .

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

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

Step 5.2

Move

\sqrt{2}

$\sqrt{2}$ .

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

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

Step 5.3

Raise

\sqrt{2}

$\sqrt{2}$ to the power of

1

$1$ .

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

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

Step 5.4

Raise

\sqrt{2}

$\sqrt{2}$ to the power of

1

$1$ .

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

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

Step 5.5

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

Step 5.6

Add

1

$1$ and

1

$1$ .

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

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

Step 5.7

Rewrite

{\sqrt{2}}^{2}

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

2

$2$ .

Tap for more steps...

Step 5.7.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{2}

$\sqrt{2}$ as

2^{\frac{1}{2}}

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

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

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

Step 5.7.2

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

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

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

Step 5.7.3

Combine

$\frac{1}{2}$ and

$2$ .

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

Step 5.7.4

Cancel the common factor of

$2$ .

Tap for more steps...

Step 5.7.4.1

Cancel the common factor.

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

Step 5.7.4.2

Rewrite the expression.

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

Step 5.7.5

Evaluate the exponent.

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

Step 6

Simplify the numerator.

Tap for more steps...

Step 6.1

Combine using the product rule for radicals.

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

Step 6.2

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:

$1.18585412 \dots$