Algebra Examples

$\sqrt{15 n^{2}} \cdot \sqrt{10 n^{3}}$

Step 1

Reorder $15$ and $n^{2}$ .

$\sqrt{n^{2} \cdot 15} \cdot \sqrt{10 n^{3}}$

Step 2

Pull terms out from under the radical.

$| n | \sqrt{15} \cdot \sqrt{10 n^{3}}$

Step 3

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

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Step 3.1

Factor out $n^{2}$ .

$| n | \sqrt{15} \cdot \sqrt{10 (n^{2} n)}$

Step 3.2

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

$| n | \sqrt{15} \cdot \sqrt{n^{2} \cdot 10 n}$

Step 3.3

Add parentheses.

$| n | \sqrt{15} \cdot \sqrt{n^{2} \cdot (10 n)}$

Step 4

Pull terms out from under the radical.

$| n | \sqrt{15} \cdot (| n | \sqrt{10 n})$

Step 5

Multiply $| n | \sqrt{15} (| n | \sqrt{10 n})$ .

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Step 5.1

To multiply absolute values, multiply the terms inside each absolute value.

$| n \cdot n | \sqrt{15} \sqrt{10 n}$

Step 5.2

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

$| n^{1} n | \sqrt{15} \sqrt{10 n}$

Step 5.3

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

$| n^{1} n^{1} | \sqrt{15} \sqrt{10 n}$

Step 5.4

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

$| n^{1 + 1} | \sqrt{15} \sqrt{10 n}$

Step 5.5

Add $1$ and $1$ .

$| n^{2} | \sqrt{15} \sqrt{10 n}$

Step 5.6

Combine using the product rule for radicals.

$| n^{2} | \sqrt{10 n \cdot 15}$

Step 5.7

Multiply $15$ by $10$ .

$| n^{2} | \sqrt{150 n}$

Step 6

Remove non-negative terms from the absolute value.

$n^{2} \sqrt{150 n}$

Step 7

Rewrite $150 n$ as $5^{2} \cdot (6 n)$ .

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Step 7.1

Factor $25$ out of $150$ .

$n^{2} \sqrt{25 (6) n}$

Step 7.2

Rewrite $25$ as $5^{2}$ .

$n^{2} \sqrt{5^{2} \cdot 6 n}$

Step 7.3

Add parentheses.

$n^{2} \sqrt{5^{2} \cdot (6 n)}$

Step 8

Pull terms out from under the radical.

$n^{2} (| 5 | \sqrt{6 n})$

Step 9

Rewrite using the commutative property of multiplication.

$| 5 | n^{2} \sqrt{6 n}$

Step 10

The absolute value is the distance between a number and zero. The distance between $0$ and $5$ is $5$ .

$5 n^{2} \sqrt{6 n}$