Basic Math Examples

$\frac{6^{3 \sqrt{54 n^{5}}}}{2^{3} \sqrt{2 n}}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $54 n^{5}$ as ${(3 n^{2})}^{2} \cdot (6 n)$ .

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Step 1.1.1

Factor $9$ out of $54$ .

$\frac{6^{3 \sqrt{9 (6) n^{5}}}}{2^{3} \sqrt{2 n}}$

Step 1.1.2

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

$\frac{6^{3 \sqrt{3^{2} \cdot 6 n^{5}}}}{2^{3} \sqrt{2 n}}$

Step 1.1.3

Factor out $n^{4}$ .

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

Step 1.1.4

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

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

Step 1.1.5

Move $6$ .

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

Step 1.1.6

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

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

Step 1.1.7

Add parentheses.

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

Step 1.2

Pull terms out from under the radical.

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

Step 1.3

Multiply $3$ by $3$ .

$\frac{6^{9 n^{2} \sqrt{6 n}}}{2^{3} \sqrt{2 n}}$

Step 2

Raise $2$ to the power of $3$ .

$\frac{6^{9 n^{2} \sqrt{6 n}}}{8 \sqrt{2 n}}$

Step 3

Multiply $\frac{6^{9 n^{2} \sqrt{6 n}}}{8 \sqrt{2 n}}$ by $\frac{\sqrt{2 n}}{\sqrt{2 n}}$ .

$\frac{6^{9 n^{2} \sqrt{6 n}}}{8 \sqrt{2 n}} \cdot \frac{\sqrt{2 n}}{\sqrt{2 n}}$

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply $\frac{6^{9 n^{2} \sqrt{6 n}}}{8 \sqrt{2 n}}$ by $\frac{\sqrt{2 n}}{\sqrt{2 n}}$ .

$\frac{6^{9 n^{2} \sqrt{6 n}} \sqrt{2 n}}{8 \sqrt{2 n} \sqrt{2 n}}$

Step 4.2

Move $\sqrt{2 n}$ .

$\frac{6^{9 n^{2} \sqrt{6 n}} \sqrt{2 n}}{8 (\sqrt{2 n} \sqrt{2 n})}$

Step 4.3

Raise $\sqrt{2 n}$ to the power of $1$ .

$\frac{6^{9 n^{2} \sqrt{6 n}} \sqrt{2 n}}{8 ({\sqrt{2 n}}^{1} \sqrt{2 n})}$

Step 4.4

Raise $\sqrt{2 n}$ to the power of $1$ .

$\frac{6^{9 n^{2} \sqrt{6 n}} \sqrt{2 n}}{8 ({\sqrt{2 n}}^{1} {\sqrt{2 n}}^{1})}$

Step 4.5

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

$\frac{6^{9 n^{2} \sqrt{6 n}} \sqrt{2 n}}{8 {\sqrt{2 n}}^{1 + 1}}$

Step 4.6

Add $1$ and $1$ .

$\frac{6^{9 n^{2} \sqrt{6 n}} \sqrt{2 n}}{8 {\sqrt{2 n}}^{2}}$

Step 4.7

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

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Step 4.7.1

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

$\frac{6^{9 n^{2} \sqrt{6 n}} \sqrt{2 n}}{8 {({(2 n)}^{\frac{1}{2}})}^{2}}$

Step 4.7.2

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

$\frac{6^{9 n^{2} \sqrt{6 n}} \sqrt{2 n}}{8 {(2 n)}^{\frac{1}{2} \cdot 2}}$

Step 4.7.3

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

$\frac{6^{9 n^{2} \sqrt{6 n}} \sqrt{2 n}}{8 {(2 n)}^{\frac{2}{2}}}$

Step 4.7.4

Cancel the common factor of $2$ .

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Step 4.7.4.1

Cancel the common factor.

$\frac{6^{9 n^{2} \sqrt{6 n}} \sqrt{2 n}}{8 {(2 n)}^{\frac{2}{2}}}$

Step 4.7.4.2

Rewrite the expression.

$\frac{6^{9 n^{2} \sqrt{6 n}} \sqrt{2 n}}{8 {(2 n)}^{1}}$

Step 4.7.5

Simplify.

$\frac{6^{9 n^{2} \sqrt{6 n}} \sqrt{2 n}}{8 (2 n)}$

Step 5

Multiply $8$ by $2$ .

$\frac{6^{9 n^{2} \sqrt{6 n}} \sqrt{2 n}}{16 n}$