Algebra Examples

$\frac{1}{5} \sqrt{5 n^{3}} \cdot \sqrt{20 n^{5}}$

Step 1

Rewrite $5 n^{3}$ as $n^{2} \cdot (5 n)$ .

Tap for more steps...

Step 1.1

Factor out $n^{2}$ .

$\frac{1}{5} \cdot \sqrt{5 (n^{2} n)} \cdot \sqrt{20 n^{5}}$

Step 1.2

Reorder $5$ and $n^{2}$ .

$\frac{1}{5} \cdot \sqrt{n^{2} \cdot 5 n} \cdot \sqrt{20 n^{5}}$

Step 1.3

Add parentheses.

$\frac{1}{5} \cdot \sqrt{n^{2} \cdot (5 n)} \cdot \sqrt{20 n^{5}}$

Step 2

Pull terms out from under the radical.

$\frac{1}{5} \cdot (n \sqrt{5 n}) \cdot \sqrt{20 n^{5}}$

Step 3

Multiply $\frac{1}{5} (n \sqrt{5 n})$ .

Tap for more steps...

Step 3.1

Combine $n$ and $\frac{1}{5}$ .

$\frac{n}{5} \sqrt{5 n} \cdot \sqrt{20 n^{5}}$

Step 3.2

Combine $\frac{n}{5}$ and $\sqrt{5 n}$ .

$\frac{n \sqrt{5 n}}{5} \cdot \sqrt{20 n^{5}}$

Step 4

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

Tap for more steps...

Step 4.1

Factor $4$ out of $20$ .

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

Step 4.2

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

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

Step 4.3

Factor out $n^{4}$ .

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

Step 4.4

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

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

Step 4.5

Move $5$ .

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

Step 4.6

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

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

Step 4.7

Add parentheses.

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

Step 5

Pull terms out from under the radical.

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

Step 6

Rewrite using the commutative property of multiplication.

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

Step 7

Combine $2$ and $\frac{n \sqrt{5 n}}{5}$ .

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

Step 8

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

Tap for more steps...

Step 8.1

Combine $n^{2}$ and $\frac{2 n \sqrt{5 n}}{5}$ .

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

Step 8.2

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

Tap for more steps...

Step 8.2.1

Move $n$ .

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

Step 8.2.2

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

Tap for more steps...

Step 8.2.2.1

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

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

Step 8.2.2.2

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

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

Step 8.2.3

Add $1$ and $2$ .

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

Step 8.3

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

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

Step 8.4

Raise $\sqrt{5 n}$ to the power of $1$ .

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

Step 8.5

Raise $\sqrt{5 n}$ to the power of $1$ .

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

Step 8.6

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

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

Step 8.7

Add $1$ and $1$ .

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

Step 9

Simplify the numerator.

Tap for more steps...

Step 9.1

Rewrite.

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

Step 9.2

Raise $\sqrt{5 n}$ to the power of $1$ .

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

Step 9.3

Raise $\sqrt{5 n}$ to the power of $1$ .

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

Step 9.4

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

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

Step 9.5

Add $1$ and $1$ .

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

Step 9.6

Remove unnecessary parentheses.

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

Step 9.7

Rewrite ${\sqrt{5 n}}^{2}$ as $5 n$ .

Tap for more steps...

Step 9.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 n}$ as ${(5 n)}^{\frac{1}{2}}$ .

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

Step 9.7.2

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

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

Step 9.7.3

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

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

Step 9.7.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 9.7.4.1

Cancel the common factor.

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

Step 9.7.4.2

Rewrite the expression.

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

Step 9.7.5

Simplify.

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

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

Step 9.8

Combine exponents.

Tap for more steps...

Step 9.8.1

Multiply $5$ by $2$ .

$\frac{n^{3} \cdot 10 n}{5}$

Step 9.8.2

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

$\frac{10 (n^{3} n^{1})}{5}$

Step 9.8.3

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

$\frac{10 n^{3 + 1}}{5}$

Step 9.8.4

Add $3$ and $1$ .

$\frac{10 n^{4}}{5}$

Step 10

Cancel the common factor of $10$ and $5$ .

Tap for more steps...

Step 10.1

Factor $5$ out of $10 n^{4}$ .

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

Step 10.2

Cancel the common factors.

Tap for more steps...

Step 10.2.1

Factor $5$ out of $5$ .

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

Step 10.2.2

Cancel the common factor.

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

Step 10.2.3

Rewrite the expression.

$\frac{2 n^{4}}{1}$

Step 10.2.4

Divide $2 n^{4}$ by $1$ .

$2 n^{4}$