Algebra Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

$\frac{3 n^{2} + {(2 n^{2})}^{\frac{1}{2}}}{\sqrt{10 n}}$

Step 1.2

Apply the product rule to $2 n^{2}$ .

$\frac{3 n^{2} + 2^{\frac{1}{2}} {(n^{2})}^{\frac{1}{2}}}{\sqrt{10 n}}$

Step 1.3

Multiply the exponents in ${(n^{2})}^{\frac{1}{2}}$ .

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Step 1.3.1

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

$\frac{3 n^{2} + 2^{\frac{1}{2}} n^{2 (\frac{1}{2})}}{\sqrt{10 n}}$

Step 1.3.2

Cancel the common factor of $2$ .

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Step 1.3.2.1

Cancel the common factor.

$\frac{3 n^{2} + 2^{\frac{1}{2}} n^{2 (\frac{1}{2})}}{\sqrt{10 n}}$

Step 1.3.2.2

Rewrite the expression.

$\frac{3 n^{2} + 2^{\frac{1}{2}} n^{1}}{\sqrt{10 n}}$

Step 1.4

Simplify.

$\frac{3 n^{2} + 2^{\frac{1}{2}} n}{\sqrt{10 n}}$

Step 1.5

Factor $n$ out of $3 n^{2} + 2^{\frac{1}{2}} n$ .

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Step 1.5.1

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

$\frac{n (3 n) + 2^{\frac{1}{2}} n}{\sqrt{10 n}}$

Step 1.5.2

Factor $n$ out of $2^{\frac{1}{2}} n$ .

$\frac{n (3 n) + n \cdot 2^{\frac{1}{2}}}{\sqrt{10 n}}$

Step 1.5.3

Factor $n$ out of $n (3 n) + n \cdot 2^{\frac{1}{2}}$ .

$\frac{n (3 n + 2^{\frac{1}{2}})}{\sqrt{10 n}}$

Step 2

Multiply $\frac{n (3 n + 2^{\frac{1}{2}})}{\sqrt{10 n}}$ by $\frac{\sqrt{10 n}}{\sqrt{10 n}}$ .

$\frac{n (3 n + 2^{\frac{1}{2}})}{\sqrt{10 n}} \cdot \frac{\sqrt{10 n}}{\sqrt{10 n}}$

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply $\frac{n (3 n + 2^{\frac{1}{2}})}{\sqrt{10 n}}$ by $\frac{\sqrt{10 n}}{\sqrt{10 n}}$ .

$\frac{n (3 n + 2^{\frac{1}{2}}) \sqrt{10 n}}{\sqrt{10 n} \sqrt{10 n}}$

Step 3.2

Raise $\sqrt{10 n}$ to the power of $1$ .

$\frac{n (3 n + 2^{\frac{1}{2}}) \sqrt{10 n}}{{\sqrt{10 n}}^{1} \sqrt{10 n}}$

Step 3.3

Raise $\sqrt{10 n}$ to the power of $1$ .

$\frac{n (3 n + 2^{\frac{1}{2}}) \sqrt{10 n}}{{\sqrt{10 n}}^{1} {\sqrt{10 n}}^{1}}$

Step 3.4

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

$\frac{n (3 n + 2^{\frac{1}{2}}) \sqrt{10 n}}{{\sqrt{10 n}}^{1 + 1}}$

Step 3.5

Add $1$ and $1$ .

$\frac{n (3 n + 2^{\frac{1}{2}}) \sqrt{10 n}}{{\sqrt{10 n}}^{2}}$

Step 3.6

Rewrite ${\sqrt{10 n}}^{2}$ as $10 n$ .

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Step 3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10 n}$ as ${(10 n)}^{\frac{1}{2}}$ .

$\frac{n (3 n + 2^{\frac{1}{2}}) \sqrt{10 n}}{{({(10 n)}^{\frac{1}{2}})}^{2}}$

Step 3.6.2

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

$\frac{n (3 n + 2^{\frac{1}{2}}) \sqrt{10 n}}{{(10 n)}^{\frac{1}{2} \cdot 2}}$

Step 3.6.3

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

$\frac{n (3 n + 2^{\frac{1}{2}}) \sqrt{10 n}}{{(10 n)}^{\frac{2}{2}}}$

Step 3.6.4

Cancel the common factor of $2$ .

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Step 3.6.4.1

Cancel the common factor.

$\frac{n (3 n + 2^{\frac{1}{2}}) \sqrt{10 n}}{{(10 n)}^{\frac{2}{2}}}$

Step 3.6.4.2

Rewrite the expression.

$\frac{n (3 n + 2^{\frac{1}{2}}) \sqrt{10 n}}{{(10 n)}^{1}}$

Step 3.6.5

Simplify.

$\frac{n (3 n + 2^{\frac{1}{2}}) \sqrt{10 n}}{10 n}$

Step 4

Cancel the common factor.

$\frac{n (3 n + 2^{\frac{1}{2}}) \sqrt{10 n}}{10 n}$

Step 5

Rewrite the expression.

$\frac{(3 n + 2^{\frac{1}{2}}) \sqrt{10 n}}{10}$

Step 6

Reorder factors in $\frac{(3 n + 2^{\frac{1}{2}}) \sqrt{10 n}}{10}$ .

$\frac{\sqrt{10 n} (3 n + 2^{\frac{1}{2}})}{10}$