Basic Math Examples

$\sqrt{10 n^{5}} \cdot \sqrt{18 n}$

Step 1

Rewrite $10 n^{5}$ as ${(n^{2})}^{2} \cdot (10 n)$ .

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Step 1.1

Factor out $n^{4}$ .

$\sqrt{10 (n^{4} n)} \cdot \sqrt{18 n}$

Step 1.2

Rewrite $n^{4}$ as ${(n^{2})}^{2}$ .

$\sqrt{10 ({(n^{2})}^{2} n)} \cdot \sqrt{18 n}$

Step 1.3

Reorder $10$ and ${(n^{2})}^{2}$ .

$\sqrt{{(n^{2})}^{2} \cdot 10 n} \cdot \sqrt{18 n}$

Step 1.4

Add parentheses.

$\sqrt{{(n^{2})}^{2} \cdot (10 n)} \cdot \sqrt{18 n}$

Step 2

Pull terms out from under the radical.

$n^{2} \sqrt{10 n} \cdot \sqrt{18 n}$

Step 3

Rewrite $18 n$ as $3^{2} \cdot (2 n)$ .

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Step 3.1

Factor $9$ out of $18$ .

$n^{2} \sqrt{10 n} \cdot \sqrt{9 (2) n}$

Step 3.2

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

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

Step 3.3

Add parentheses.

$n^{2} \sqrt{10 n} \cdot \sqrt{3^{2} \cdot (2 n)}$

Step 4

Pull terms out from under the radical.

$n^{2} \sqrt{10 n} \cdot (3 \sqrt{2 n})$

Step 5

Rewrite using the commutative property of multiplication.

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

Step 6

Multiply $3 n^{2} \sqrt{10 n} \sqrt{2 n}$ .

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Step 6.1

Combine using the product rule for radicals.

$3 n^{2} \sqrt{2 n (10 n)}$

Step 6.2

Multiply $10$ by $2$ .

$3 n^{2} \sqrt{20 n \cdot n}$

Step 6.3

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

$3 n^{2} \sqrt{20 (n^{1} n)}$

Step 6.4

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

$3 n^{2} \sqrt{20 (n^{1} n^{1})}$

Step 6.5

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

$3 n^{2} \sqrt{20 n^{1 + 1}}$

Step 6.6

Add $1$ and $1$ .

$3 n^{2} \sqrt{20 n^{2}}$

Step 7

Rewrite $20 n^{2}$ as ${(2 n)}^{2} \cdot 5$ .

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Step 7.1

Factor $4$ out of $20$ .

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

Step 7.2

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

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

Step 7.3

Move $5$ .

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

Step 7.4

Rewrite $2^{2} n^{2}$ as ${(2 n)}^{2}$ .

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

Step 8

Pull terms out from under the radical.

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

Step 9

Multiply $n^{2}$ by $n$ by adding the exponents.

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Step 9.1

Move $n$ .

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

Step 9.2

Multiply $n$ by $n^{2}$ .

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Step 9.2.1

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

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

Step 9.2.2

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

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

Step 9.3

Add $1$ and $2$ .

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

Step 10

Multiply $2$ by $3$ .

$6 n^{3} \sqrt{5}$