Precalculus Examples

$\sqrt{2 n - 88} = \sqrt{\frac{n}{6}}$

Step 1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{2 n - 88}}^{2} = {\sqrt{\frac{n}{6}}}^{2}$

Step 2

Simplify each side of the equation.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 n - 88}$ as ${(2 n - 88)}^{\frac{1}{2}}$ .

${({(2 n - 88)}^{\frac{1}{2}})}^{2} = {\sqrt{\frac{n}{6}}}^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(2 n - 88)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(2 n - 88)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

${(2 n - 88)}^{\frac{1}{2} \cdot 2} = {\sqrt{\frac{n}{6}}}^{2}$

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

${(2 n - 88)}^{\frac{1}{2} \cdot 2} = {\sqrt{\frac{n}{6}}}^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(2 n - 88)}^{1} = {\sqrt{\frac{n}{6}}}^{2}$

Step 2.2.1.2

Simplify.

$2 n - 88 = {\sqrt{\frac{n}{6}}}^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify ${\sqrt{\frac{n}{6}}}^{2}$ .

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Step 2.3.1.1

Rewrite $\sqrt{\frac{n}{6}}$ as $\frac{\sqrt{n}}{\sqrt{6}}$ .

$2 n - 88 = {(\frac{\sqrt{n}}{\sqrt{6}})}^{2}$

Step 2.3.1.2

Multiply $\frac{\sqrt{n}}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$2 n - 88 = {(\frac{\sqrt{n}}{\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}})}^{2}$

Step 2.3.1.3

Simplify terms.

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Step 2.3.1.3.1

Combine and simplify the denominator.

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Step 2.3.1.3.1.1

Multiply $\frac{\sqrt{n}}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$2 n - 88 = {(\frac{\sqrt{n} \sqrt{6}}{\sqrt{6} \sqrt{6}})}^{2}$

Step 2.3.1.3.1.2

Raise $\sqrt{6}$ to the power of $1$ .

$2 n - 88 = {(\frac{\sqrt{n} \sqrt{6}}{{\sqrt{6}}^{1} \sqrt{6}})}^{2}$

Step 2.3.1.3.1.3

Raise $\sqrt{6}$ to the power of $1$ .

$2 n - 88 = {(\frac{\sqrt{n} \sqrt{6}}{{\sqrt{6}}^{1} {\sqrt{6}}^{1}})}^{2}$

Step 2.3.1.3.1.4

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

$2 n - 88 = {(\frac{\sqrt{n} \sqrt{6}}{{\sqrt{6}}^{1 + 1}})}^{2}$

Step 2.3.1.3.1.5

Add $1$ and $1$ .

$2 n - 88 = {(\frac{\sqrt{n} \sqrt{6}}{{\sqrt{6}}^{2}})}^{2}$

Step 2.3.1.3.1.6

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 2.3.1.3.1.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$2 n - 88 = {(\frac{\sqrt{n} \sqrt{6}}{{(6^{\frac{1}{2}})}^{2}})}^{2}$

Step 2.3.1.3.1.6.2

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

$2 n - 88 = {(\frac{\sqrt{n} \sqrt{6}}{6^{\frac{1}{2} \cdot 2}})}^{2}$

Step 2.3.1.3.1.6.3

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

$2 n - 88 = {(\frac{\sqrt{n} \sqrt{6}}{6^{\frac{2}{2}}})}^{2}$

Step 2.3.1.3.1.6.4

Cancel the common factor of $2$ .

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Step 2.3.1.3.1.6.4.1

Cancel the common factor.

$2 n - 88 = {(\frac{\sqrt{n} \sqrt{6}}{6^{\frac{2}{2}}})}^{2}$

Step 2.3.1.3.1.6.4.2

Rewrite the expression.

$2 n - 88 = {(\frac{\sqrt{n} \sqrt{6}}{6^{1}})}^{2}$

Step 2.3.1.3.1.6.5

Evaluate the exponent.

$2 n - 88 = {(\frac{\sqrt{n} \sqrt{6}}{6})}^{2}$

Step 2.3.1.3.2

Combine using the product rule for radicals.

$2 n - 88 = {(\frac{\sqrt{n \cdot 6}}{6})}^{2}$

Step 2.3.1.3.3

Apply the product rule to $\frac{\sqrt{n \cdot 6}}{6}$ .

$2 n - 88 = \frac{{\sqrt{n \cdot 6}}^{2}}{6^{2}}$

Step 2.3.1.4

Simplify the numerator.

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Step 2.3.1.4.1

Rewrite ${\sqrt{n \cdot 6}}^{2}$ as $n \cdot 6$ .

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Step 2.3.1.4.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{n \cdot 6}$ as ${(n \cdot 6)}^{\frac{1}{2}}$ .

$2 n - 88 = \frac{{({(n \cdot 6)}^{\frac{1}{2}})}^{2}}{6^{2}}$

Step 2.3.1.4.1.2

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

$2 n - 88 = \frac{{(n \cdot 6)}^{\frac{1}{2} \cdot 2}}{6^{2}}$

Step 2.3.1.4.1.3

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

$2 n - 88 = \frac{{(n \cdot 6)}^{\frac{2}{2}}}{6^{2}}$

Step 2.3.1.4.1.4

Cancel the common factor of $2$ .

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Step 2.3.1.4.1.4.1

Cancel the common factor.

$2 n - 88 = \frac{{(n \cdot 6)}^{\frac{2}{2}}}{6^{2}}$

Step 2.3.1.4.1.4.2

Rewrite the expression.

$2 n - 88 = \frac{{(n \cdot 6)}^{1}}{6^{2}}$

Step 2.3.1.4.1.5

Simplify.

$2 n - 88 = \frac{n \cdot 6}{6^{2}}$

Step 2.3.1.4.2

Move $6$ to the left of $n$ .

$2 n - 88 = \frac{6 n}{6^{2}}$

Step 2.3.1.5

Reduce the expression by cancelling the common factors.

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Step 2.3.1.5.1

Raise $6$ to the power of $2$ .

$2 n - 88 = \frac{6 n}{36}$

Step 2.3.1.5.2

Cancel the common factor of $6$ and $36$ .

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Step 2.3.1.5.2.1

Factor $6$ out of $6 n$ .

$2 n - 88 = \frac{6 (n)}{36}$

Step 2.3.1.5.2.2

Cancel the common factors.

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Step 2.3.1.5.2.2.1

Factor $6$ out of $36$ .

$2 n - 88 = \frac{6 n}{6 \cdot 6}$

Step 2.3.1.5.2.2.2

Cancel the common factor.

$2 n - 88 = \frac{6 n}{6 \cdot 6}$

Step 2.3.1.5.2.2.3

Rewrite the expression.

$2 n - 88 = \frac{n}{6}$

Step 3

Solve for $n$ .

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Step 3.1

Multiply both sides by $6$ .

$(2 n - 88) \cdot 6 = \frac{n}{6} \cdot 6$

Step 3.2

Simplify.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify $(2 n - 88) \cdot 6$ .

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Step 3.2.1.1.1

Apply the distributive property.

$2 n \cdot 6 - 88 \cdot 6 = \frac{n}{6} \cdot 6$

Step 3.2.1.1.2

Multiply.

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Step 3.2.1.1.2.1

Multiply $6$ by $2$ .

$12 n - 88 \cdot 6 = \frac{n}{6} \cdot 6$

Step 3.2.1.1.2.2

Multiply $- 88$ by $6$ .

$12 n - 528 = \frac{n}{6} \cdot 6$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Cancel the common factor of $6$ .

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Step 3.2.2.1.1

Cancel the common factor.

$12 n - 528 = \frac{n}{6} \cdot 6$

Step 3.2.2.1.2

Rewrite the expression.

$12 n - 528 = n$

Step 3.3

Solve for $n$ .

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Step 3.3.1

Move all terms containing $n$ to the left side of the equation.

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Step 3.3.1.1

Subtract $n$ from both sides of the equation.

$12 n - 528 - n = 0$

Step 3.3.1.2

Subtract $n$ from $12 n$ .

$11 n - 528 = 0$

Step 3.3.2

Add $528$ to both sides of the equation.

$11 n = 528$

Step 3.3.3

Divide each term in $11 n = 528$ by $11$ and simplify.

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Step 3.3.3.1

Divide each term in $11 n = 528$ by $11$ .

$\frac{11 n}{11} = \frac{528}{11}$

Step 3.3.3.2

Simplify the left side.

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Step 3.3.3.2.1

Cancel the common factor of $11$ .

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Step 3.3.3.2.1.1

Cancel the common factor.

$\frac{11 n}{11} = \frac{528}{11}$

Step 3.3.3.2.1.2

Divide $n$ by $1$ .

$n = \frac{528}{11}$

Step 3.3.3.3

Simplify the right side.

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Step 3.3.3.3.1

Divide $528$ by $11$ .

$n = 48$