Basic Math Examples

$\frac{\sqrt{9 p^{15}}}{\sqrt{10 r^{3}}}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $9 p^{15}$ as ${(3 p^{7})}^{2} p$ .

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Step 1.1.1

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

$\frac{\sqrt{3^{2} p^{15}}}{\sqrt{10 r^{3}}}$

Step 1.1.2

Factor out $p^{14}$ .

$\frac{\sqrt{3^{2} (p^{14} p)}}{\sqrt{10 r^{3}}}$

Step 1.1.3

Rewrite $p^{14}$ as ${(p^{7})}^{2}$ .

$\frac{\sqrt{3^{2} ({(p^{7})}^{2} p)}}{\sqrt{10 r^{3}}}$

Step 1.1.4

Rewrite $3^{2} {(p^{7})}^{2}$ as ${(3 p^{7})}^{2}$ .

$\frac{\sqrt{{(3 p^{7})}^{2} p}}{\sqrt{10 r^{3}}}$

Step 1.2

Pull terms out from under the radical.

$\frac{3 p^{7} \sqrt{p}}{\sqrt{10 r^{3}}}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $10 r^{3}$ as $r^{2} \cdot (10 r)$ .

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Step 2.1.1

Factor out $r^{2}$ .

$\frac{3 p^{7} \sqrt{p}}{\sqrt{10 (r^{2} r)}}$

Step 2.1.2

Reorder $10$ and $r^{2}$ .

$\frac{3 p^{7} \sqrt{p}}{\sqrt{r^{2} \cdot 10 r}}$

Step 2.1.3

Add parentheses.

$\frac{3 p^{7} \sqrt{p}}{\sqrt{r^{2} \cdot (10 r)}}$

Step 2.2

Pull terms out from under the radical.

$\frac{3 p^{7} \sqrt{p}}{r \sqrt{10 r}}$

Step 3

Multiply $\frac{3 p^{7} \sqrt{p}}{r \sqrt{10 r}}$ by $\frac{\sqrt{10 r}}{\sqrt{10 r}}$ .

$\frac{3 p^{7} \sqrt{p}}{r \sqrt{10 r}} \cdot \frac{\sqrt{10 r}}{\sqrt{10 r}}$

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply $\frac{3 p^{7} \sqrt{p}}{r \sqrt{10 r}}$ by $\frac{\sqrt{10 r}}{\sqrt{10 r}}$ .

$\frac{3 p^{7} \sqrt{p} \sqrt{10 r}}{r \sqrt{10 r} \sqrt{10 r}}$

Step 4.2

Move $\sqrt{10 r}$ .

$\frac{3 p^{7} \sqrt{p} \sqrt{10 r}}{r (\sqrt{10 r} \sqrt{10 r})}$

Step 4.3

Raise $\sqrt{10 r}$ to the power of $1$ .

$\frac{3 p^{7} \sqrt{p} \sqrt{10 r}}{r ({\sqrt{10 r}}^{1} \sqrt{10 r})}$

Step 4.4

Raise $\sqrt{10 r}$ to the power of $1$ .

$\frac{3 p^{7} \sqrt{p} \sqrt{10 r}}{r ({\sqrt{10 r}}^{1} {\sqrt{10 r}}^{1})}$

Step 4.5

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

$\frac{3 p^{7} \sqrt{p} \sqrt{10 r}}{r {\sqrt{10 r}}^{1 + 1}}$

Step 4.6

Add $1$ and $1$ .

$\frac{3 p^{7} \sqrt{p} \sqrt{10 r}}{r {\sqrt{10 r}}^{2}}$

Step 4.7

Rewrite ${\sqrt{10 r}}^{2}$ as $10 r$ .

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Step 4.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10 r}$ as ${(10 r)}^{\frac{1}{2}}$ .

$\frac{3 p^{7} \sqrt{p} \sqrt{10 r}}{r {({(10 r)}^{\frac{1}{2}})}^{2}}$

Step 4.7.2

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

$\frac{3 p^{7} \sqrt{p} \sqrt{10 r}}{r {(10 r)}^{\frac{1}{2} \cdot 2}}$

Step 4.7.3

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

$\frac{3 p^{7} \sqrt{p} \sqrt{10 r}}{r {(10 r)}^{\frac{2}{2}}}$

Step 4.7.4

Cancel the common factor of $2$ .

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Step 4.7.4.1

Cancel the common factor.

$\frac{3 p^{7} \sqrt{p} \sqrt{10 r}}{r {(10 r)}^{\frac{2}{2}}}$

Step 4.7.4.2

Rewrite the expression.

$\frac{3 p^{7} \sqrt{p} \sqrt{10 r}}{r {(10 r)}^{1}}$

Step 4.7.5

Simplify.

$\frac{3 p^{7} \sqrt{p} \sqrt{10 r}}{r (10 r)}$

Step 5

Multiply $r$ by $r$ by adding the exponents.

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Step 5.1

Move $r$ .

$\frac{3 p^{7} \sqrt{p} \sqrt{10 r}}{r \cdot r \cdot 10}$

Step 5.2

Multiply $r$ by $r$ .

$\frac{3 p^{7} \sqrt{p} \sqrt{10 r}}{r^{2} \cdot 10}$

Step 6

Combine using the product rule for radicals.

$\frac{3 p^{7} \sqrt{10 r p}}{r^{2} \cdot 10}$

Step 7

Move $10$ to the left of $r^{2}$ .

$\frac{3 p^{7} \sqrt{10 r p}}{10 r^{2}}$