Basic Math Examples

$\frac{(5 n \sqrt{6 n}) (4 n^{4} \sqrt{3 n^{3}})}{2 n \sqrt{2 n^{3}}}$

Step 1

Multiply $n$ by $n^{4}$ by adding the exponents.

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Step 1.1

Move $n^{4}$ .

$\frac{5 (n^{4} n) \sqrt{6 n} (4 \sqrt{3 n^{3}})}{2 n \sqrt{2 n^{3}}}$

Step 1.2

Multiply $n^{4}$ by $n$ .

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Step 1.2.1

Raise $n$ to the power of $1$ .

$\frac{5 (n^{4} n^{1}) \sqrt{6 n} (4 \sqrt{3 n^{3}})}{2 n \sqrt{2 n^{3}}}$

Step 1.2.2

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

$\frac{5 n^{4 + 1} \sqrt{6 n} (4 \sqrt{3 n^{3}})}{2 n \sqrt{2 n^{3}}}$

Step 1.3

Add $4$ and $1$ .

$\frac{5 n^{5} \sqrt{6 n} (4 \sqrt{3 n^{3}})}{2 n \sqrt{2 n^{3}}}$

Step 2

Cancel the common factor of $n^{5}$ and $n$ .

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Step 2.1

Factor $n$ out of $5 n^{5} \sqrt{6 n} (4 \sqrt{3 n^{3}})$ .

$\frac{n (5 n^{4} \sqrt{6 n} (4 \sqrt{3 n^{3}}))}{2 n \sqrt{2 n^{3}}}$

Step 2.2

Cancel the common factors.

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Step 2.2.1

Factor $n$ out of $2 n \sqrt{2 n^{3}}$ .

$\frac{n (5 n^{4} \sqrt{6 n} (4 \sqrt{3 n^{3}}))}{n (2 \sqrt{2 n^{3}})}$

Step 2.2.2

Cancel the common factor.

$\frac{n (5 n^{4} \sqrt{6 n} (4 \sqrt{3 n^{3}}))}{n (2 \sqrt{2 n^{3}})}$

Step 2.2.3

Rewrite the expression.

$\frac{5 n^{4} \sqrt{6 n} (4 \sqrt{3 n^{3}})}{2 \sqrt{2 n^{3}}}$

Step 3

Cancel the common factor of $4$ and $2$ .

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Step 3.1

Factor $2$ out of $5 n^{4} \sqrt{6 n} (4 \sqrt{3 n^{3}})$ .

$\frac{2 (5 n^{4} \sqrt{6 n} (2 \sqrt{3 n^{3}}))}{2 \sqrt{2 n^{3}}}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $2$ out of $2 \sqrt{2 n^{3}}$ .

$\frac{2 (5 n^{4} \sqrt{6 n} (2 \sqrt{3 n^{3}}))}{2 (\sqrt{2 n^{3}})}$

Step 3.2.2

Cancel the common factor.

$\frac{2 (5 n^{4} \sqrt{6 n} (2 \sqrt{3 n^{3}}))}{2 \sqrt{2 n^{3}}}$

Step 3.2.3

Rewrite the expression.

$\frac{5 n^{4} \sqrt{6 n} (2 \sqrt{3 n^{3}})}{\sqrt{2 n^{3}}}$

Step 4

Combine $\sqrt{6 n}$ and $\sqrt{2 n^{3}}$ into a single radical.

$5 n^{4} \sqrt{\frac{6 n}{2 n^{3}}} (2 \sqrt{3 n^{3}})$

Step 5

Reduce the expression $\frac{6 n}{2 n^{3}}$ by cancelling the common factors.

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Step 5.1

Factor $2$ out of $6 n$ .

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

Step 5.2

Factor $2$ out of $2 n^{3}$ .

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

Step 5.3

Cancel the common factor.

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

Step 5.4

Rewrite the expression.

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

Step 6

Cancel the common factor of $n$ and $n^{3}$ .

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Step 6.1

Factor $n$ out of $3 n$ .

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

Step 6.2

Cancel the common factors.

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Step 6.2.1

Factor $n$ out of $n^{3}$ .

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

Step 6.2.2

Cancel the common factor.

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

Step 6.2.3

Rewrite the expression.

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

Step 7

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

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Step 7.1

Factor the perfect power $1^{2}$ out of $3$ .

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

Step 7.2

Factor the perfect power $n^{2}$ out of $n^{2}$ .

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

Step 7.3

Rearrange the fraction $\frac{1^{2} \cdot 3}{n^{2} \cdot 1}$ .

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

Step 8

Pull terms out from under the radical.

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

Step 9

Combine $\frac{1}{n}$ and $\sqrt{3}$ .

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

Step 10

Cancel the common factor of $n$ .

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Step 10.1

Factor $n$ out of $5 n^{4}$ .

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

Step 10.2

Cancel the common factor.

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

Step 10.3

Rewrite the expression.

$5 n^{3} \sqrt{3} (2 \sqrt{3 n^{3}})$

Step 11

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

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Step 11.1

Factor out $n^{2}$ .

$5 n^{3} \sqrt{3} (2 \sqrt{3 (n^{2} n)})$

Step 11.2

Reorder $3$ and $n^{2}$ .

$5 n^{3} \sqrt{3} (2 \sqrt{n^{2} \cdot 3 n})$

Step 11.3

Add parentheses.

$5 n^{3} \sqrt{3} (2 \sqrt{n^{2} \cdot (3 n)})$

Step 12

Pull terms out from under the radical.

$5 n^{3} \sqrt{3} (2 (n \sqrt{3 n}))$

Step 13

Multiply $n^{3}$ by $n$ by adding the exponents.

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Step 13.1

Move $n$ .

$5 (n \cdot n^{3}) \sqrt{3} (2 (\sqrt{3 n}))$

Step 13.2

Multiply $n$ by $n^{3}$ .

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Step 13.2.1

Raise $n$ to the power of $1$ .

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

Step 13.2.2

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

$5 n^{1 + 3} \sqrt{3} (2 (\sqrt{3 n}))$

Step 13.3

Add $1$ and $3$ .

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

Step 14

Multiply $5 n^{4} \sqrt{3} (2 \sqrt{3 n})$ .

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Step 14.1

Multiply $2$ by $5$ .

$10 n^{4} \sqrt{3} \sqrt{3 n}$

Step 14.2

Combine using the product rule for radicals.

$10 n^{4} \sqrt{3 n \cdot 3}$

Step 14.3

Multiply $3$ by $3$ .

$10 n^{4} \sqrt{9 n}$

Step 15

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

$10 n^{4} \sqrt{3^{2} n}$

Step 16

Pull terms out from under the radical.

$10 n^{4} (3 \sqrt{n})$

Step 17

Multiply $3$ by $10$ .

$30 n^{4} \sqrt{n}$