Basic Math Examples

$\sqrt{\frac{45}{7 y}}$

Step 1

Rewrite $\sqrt{\frac{45}{7 y}}$ as $\frac{\sqrt{45}}{\sqrt{7 y}}$ .

$\frac{\sqrt{45}}{\sqrt{7 y}}$

Step 2

Simplify the numerator.

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Step 2.1

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

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Step 2.1.1

Factor $9$ out of $45$ .

$\frac{\sqrt{9 (5)}}{\sqrt{7 y}}$

Step 2.1.2

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

$\frac{\sqrt{3^{2} \cdot 5}}{\sqrt{7 y}}$

Step 2.2

Pull terms out from under the radical.

$\frac{3 \sqrt{5}}{\sqrt{7 y}}$

Step 3

Multiply $\frac{3 \sqrt{5}}{\sqrt{7 y}}$ by $\frac{\sqrt{7 y}}{\sqrt{7 y}}$ .

$\frac{3 \sqrt{5}}{\sqrt{7 y}} \cdot \frac{\sqrt{7 y}}{\sqrt{7 y}}$

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply $\frac{3 \sqrt{5}}{\sqrt{7 y}}$ by $\frac{\sqrt{7 y}}{\sqrt{7 y}}$ .

$\frac{3 \sqrt{5} \sqrt{7 y}}{\sqrt{7 y} \sqrt{7 y}}$

Step 4.2

Raise $\sqrt{7 y}$ to the power of $1$ .

$\frac{3 \sqrt{5} \sqrt{7 y}}{{\sqrt{7 y}}^{1} \sqrt{7 y}}$

Step 4.3

Raise $\sqrt{7 y}$ to the power of $1$ .

$\frac{3 \sqrt{5} \sqrt{7 y}}{{\sqrt{7 y}}^{1} {\sqrt{7 y}}^{1}}$

Step 4.4

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

$\frac{3 \sqrt{5} \sqrt{7 y}}{{\sqrt{7 y}}^{1 + 1}}$

Step 4.5

Add $1$ and $1$ .

$\frac{3 \sqrt{5} \sqrt{7 y}}{{\sqrt{7 y}}^{2}}$

Step 4.6

Rewrite ${\sqrt{7 y}}^{2}$ as $7 y$ .

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Step 4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7 y}$ as ${(7 y)}^{\frac{1}{2}}$ .

$\frac{3 \sqrt{5} \sqrt{7 y}}{{({(7 y)}^{\frac{1}{2}})}^{2}}$

Step 4.6.2

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

$\frac{3 \sqrt{5} \sqrt{7 y}}{{(7 y)}^{\frac{1}{2} \cdot 2}}$

Step 4.6.3

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

$\frac{3 \sqrt{5} \sqrt{7 y}}{{(7 y)}^{\frac{2}{2}}}$

Step 4.6.4

Cancel the common factor of $2$ .

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Step 4.6.4.1

Cancel the common factor.

$\frac{3 \sqrt{5} \sqrt{7 y}}{{(7 y)}^{\frac{2}{2}}}$

Step 4.6.4.2

Rewrite the expression.

$\frac{3 \sqrt{5} \sqrt{7 y}}{{(7 y)}^{1}}$

Step 4.6.5

Simplify.

$\frac{3 \sqrt{5} \sqrt{7 y}}{7 y}$

Step 5

Simplify the numerator.

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Step 5.1

Combine using the product rule for radicals.

$\frac{3 \sqrt{7 y \cdot 5}}{7 y}$

Step 5.2

Multiply $5$ by $7$ .

$\frac{3 \sqrt{35 y}}{7 y}$