Precalculus Examples

$a (n) = \frac{4}{n^{2}}$

Step 1

Find the LCD of the terms in the equation.

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Step 1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, n^{2}$

Step 1.2

The LCM of one and any expression is the expression.

$n^{2}$

Step 2

Multiply each term in $a n = \frac{4}{n^{2}}$ by $n^{2}$ to eliminate the fractions.

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Step 2.1

Multiply each term in $a n = \frac{4}{n^{2}}$ by $n^{2}$ .

$a n \cdot n^{2} = \frac{4}{n^{2}} n^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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Step 2.2.1.1

Move $n^{2}$ .

$a (n^{2} n) = \frac{4}{n^{2}} n^{2}$

Step 2.2.1.2

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

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Step 2.2.1.2.1

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

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

Step 2.2.1.2.2

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

$a n^{2 + 1} = \frac{4}{n^{2}} n^{2}$

Step 2.2.1.3

Add $2$ and $1$ .

$a n^{3} = \frac{4}{n^{2}} n^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Cancel the common factor of $n^{2}$ .

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Step 2.3.1.1

Cancel the common factor.

$a n^{3} = \frac{4}{n^{2}} n^{2}$

Step 2.3.1.2

Rewrite the expression.

$a n^{3} = 4$

Step 3

Solve the equation.

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Step 3.1

Divide each term in $a n^{3} = 4$ by $a$ and simplify.

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Step 3.1.1

Divide each term in $a n^{3} = 4$ by $a$ .

$\frac{a n^{3}}{a} = \frac{4}{a}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor of $a$ .

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Step 3.1.2.1.1

Cancel the common factor.

$\frac{a n^{3}}{a} = \frac{4}{a}$

Step 3.1.2.1.2

Divide $n^{3}$ by $1$ .

$n^{3} = \frac{4}{a}$

Step 3.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$n = \sqrt[3]{\frac{4}{a}}$

Step 3.3

Simplify $\sqrt[3]{\frac{4}{a}}$ .

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Step 3.3.1

Rewrite $\sqrt[3]{\frac{4}{a}}$ as $\frac{\sqrt[3]{4}}{\sqrt[3]{a}}$ .

$n = \frac{\sqrt[3]{4}}{\sqrt[3]{a}}$

Step 3.3.2

Multiply $\frac{\sqrt[3]{4}}{\sqrt[3]{a}}$ by $\frac{{\sqrt[3]{a}}^{2}}{{\sqrt[3]{a}}^{2}}$ .

$n = \frac{\sqrt[3]{4}}{\sqrt[3]{a}} \cdot \frac{{\sqrt[3]{a}}^{2}}{{\sqrt[3]{a}}^{2}}$

Step 3.3.3

Combine and simplify the denominator.

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Step 3.3.3.1

Multiply $\frac{\sqrt[3]{4}}{\sqrt[3]{a}}$ by $\frac{{\sqrt[3]{a}}^{2}}{{\sqrt[3]{a}}^{2}}$ .

$n = \frac{\sqrt[3]{4} {\sqrt[3]{a}}^{2}}{\sqrt[3]{a} {\sqrt[3]{a}}^{2}}$

Step 3.3.3.2

Raise $\sqrt[3]{a}$ to the power of $1$ .

$n = \frac{\sqrt[3]{4} {\sqrt[3]{a}}^{2}}{{\sqrt[3]{a}}^{1} {\sqrt[3]{a}}^{2}}$

Step 3.3.3.3

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

$n = \frac{\sqrt[3]{4} {\sqrt[3]{a}}^{2}}{{\sqrt[3]{a}}^{1 + 2}}$

Step 3.3.3.4

Add $1$ and $2$ .

$n = \frac{\sqrt[3]{4} {\sqrt[3]{a}}^{2}}{{\sqrt[3]{a}}^{3}}$

Step 3.3.3.5

Rewrite ${\sqrt[3]{a}}^{3}$ as $a$ .

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Step 3.3.3.5.1

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

$n = \frac{\sqrt[3]{4} {\sqrt[3]{a}}^{2}}{{(a^{\frac{1}{3}})}^{3}}$

Step 3.3.3.5.2

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

$n = \frac{\sqrt[3]{4} {\sqrt[3]{a}}^{2}}{a^{\frac{1}{3} \cdot 3}}$

Step 3.3.3.5.3

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

$n = \frac{\sqrt[3]{4} {\sqrt[3]{a}}^{2}}{a^{\frac{3}{3}}}$

Step 3.3.3.5.4

Cancel the common factor of $3$ .

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Step 3.3.3.5.4.1

Cancel the common factor.

$n = \frac{\sqrt[3]{4} {\sqrt[3]{a}}^{2}}{a^{\frac{3}{3}}}$

Step 3.3.3.5.4.2

Rewrite the expression.

$n = \frac{\sqrt[3]{4} {\sqrt[3]{a}}^{2}}{a^{1}}$

Step 3.3.3.5.5

Simplify.

$n = \frac{\sqrt[3]{4} {\sqrt[3]{a}}^{2}}{a}$

Step 3.3.4

Rewrite ${\sqrt[3]{a}}^{2}$ as $\sqrt[3]{a^{2}}$ .

$n = \frac{\sqrt[3]{4} \sqrt[3]{a^{2}}}{a}$

Step 3.3.5

Combine using the product rule for radicals.

$n = \frac{\sqrt[3]{4 a^{2}}}{a}$