Algebra Examples

$- \frac{4 n^{2}}{\sqrt{3 n^{4}} + \sqrt{3 n^{3}}}$

Step 1

Simplify the denominator.

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Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 n^{4}}$ as ${(3 n^{4})}^{\frac{1}{2}}$ .

$- \frac{4 n^{2}}{{(3 n^{4})}^{\frac{1}{2}} + \sqrt{3 n^{3}}}$

Step 1.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 n^{3}}$ as ${(3 n^{3})}^{\frac{1}{2}}$ .

$- \frac{4 n^{2}}{{(3 n^{4})}^{\frac{1}{2}} + {(3 n^{3})}^{\frac{1}{2}}}$

Step 1.3

Apply the product rule to $3 n^{4}$ .

$- \frac{4 n^{2}}{3^{\frac{1}{2}} {(n^{4})}^{\frac{1}{2}} + {(3 n^{3})}^{\frac{1}{2}}}$

Step 1.4

Multiply the exponents in ${(n^{4})}^{\frac{1}{2}}$ .

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Step 1.4.1

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

$- \frac{4 n^{2}}{3^{\frac{1}{2}} n^{4 (\frac{1}{2})} + {(3 n^{3})}^{\frac{1}{2}}}$

Step 1.4.2

Cancel the common factor of $2$ .

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Step 1.4.2.1

Factor $2$ out of $4$ .

$- \frac{4 n^{2}}{3^{\frac{1}{2}} n^{2 (2) \frac{1}{2}} + {(3 n^{3})}^{\frac{1}{2}}}$

Step 1.4.2.2

Cancel the common factor.

$- \frac{4 n^{2}}{3^{\frac{1}{2}} n^{2 \cdot 2 \frac{1}{2}} + {(3 n^{3})}^{\frac{1}{2}}}$

Step 1.4.2.3

Rewrite the expression.

$- \frac{4 n^{2}}{3^{\frac{1}{2}} n^{2} + {(3 n^{3})}^{\frac{1}{2}}}$

Step 1.5

Apply the product rule to $3 n^{3}$ .

$- \frac{4 n^{2}}{3^{\frac{1}{2}} n^{2} + 3^{\frac{1}{2}} {(n^{3})}^{\frac{1}{2}}}$

Step 1.6

Multiply the exponents in ${(n^{3})}^{\frac{1}{2}}$ .

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Step 1.6.1

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

$- \frac{4 n^{2}}{3^{\frac{1}{2}} n^{2} + 3^{\frac{1}{2}} n^{3 (\frac{1}{2})}}$

Step 1.6.2

Combine $3$ and $\frac{1}{2}$ .

$- \frac{4 n^{2}}{3^{\frac{1}{2}} n^{2} + 3^{\frac{1}{2}} n^{\frac{3}{2}}}$

Step 1.7

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

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Step 1.7.1

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

$- \frac{4 n^{2}}{3^{\frac{1}{2}} n^{\frac{3}{2}} (n^{\frac{1}{2}}) + 3^{\frac{1}{2}} n^{\frac{3}{2}}}$

Step 1.7.2

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

$- \frac{4 n^{2}}{3^{\frac{1}{2}} n^{\frac{3}{2}} (n^{\frac{1}{2}}) + 3^{\frac{1}{2}} n^{\frac{3}{2}} (1)}$

Step 1.7.3

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

$- \frac{4 n^{2}}{3^{\frac{1}{2}} n^{\frac{3}{2}} (n^{\frac{1}{2}} + 1)}$

Step 2

Move $n^{\frac{3}{2}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$- \frac{4 n^{2} n^{- \frac{3}{2}}}{3^{\frac{1}{2}} (n^{\frac{1}{2}} + 1)}$

Step 3

Multiply $n^{2}$ by $n^{- \frac{3}{2}}$ by adding the exponents.

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Step 3.1

Move $n^{- \frac{3}{2}}$ .

$- \frac{4 (n^{- \frac{3}{2}} n^{2})}{3^{\frac{1}{2}} (n^{\frac{1}{2}} + 1)}$

Step 3.2

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

$- \frac{4 n^{- \frac{3}{2} + 2}}{3^{\frac{1}{2}} (n^{\frac{1}{2}} + 1)}$

Step 3.3

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{4 n^{- \frac{3}{2} + 2 \cdot \frac{2}{2}}}{3^{\frac{1}{2}} (n^{\frac{1}{2}} + 1)}$

Step 3.4

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

$- \frac{4 n^{- \frac{3}{2} + \frac{2 \cdot 2}{2}}}{3^{\frac{1}{2}} (n^{\frac{1}{2}} + 1)}$

Step 3.5

Combine the numerators over the common denominator.

$- \frac{4 n^{\frac{- 3 + 2 \cdot 2}{2}}}{3^{\frac{1}{2}} (n^{\frac{1}{2}} + 1)}$

Step 3.6

Simplify the numerator.

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Step 3.6.1

Multiply $2$ by $2$ .

$- \frac{4 n^{\frac{- 3 + 4}{2}}}{3^{\frac{1}{2}} (n^{\frac{1}{2}} + 1)}$

Step 3.6.2

Add $- 3$ and $4$ .

$- \frac{4 n^{\frac{1}{2}}}{3^{\frac{1}{2}} (n^{\frac{1}{2}} + 1)}$