Precalculus Examples

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

Step 1

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

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

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 $\sqrt{n}$ as $n^{\frac{1}{2}}$ .

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

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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Step 2.2.1.1

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

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Step 2.2.1.1.1

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

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

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.

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

Step 2.2.1.1.2.2

Rewrite the expression.

$n^{1} = {\sqrt{2 n - 6}}^{2}$

Step 2.2.1.2

Simplify.

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

Step 2.3

Simplify the right side.

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Step 2.3.1

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

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Step 2.3.1.1

Factor $2$ out of $2 n - 6$ .

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Step 2.3.1.1.1

Factor $2$ out of $2 n$ .

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

Step 2.3.1.1.2

Factor $2$ out of $- 6$ .

$n = {\sqrt{2 n + 2 \cdot - 3}}^{2}$

Step 2.3.1.1.3

Factor $2$ out of $2 n + 2 \cdot - 3$ .

$n = {\sqrt{2 (n - 3)}}^{2}$

Step 2.3.1.2

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

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Step 2.3.1.2.1

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

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

Step 2.3.1.2.2

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

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

Step 2.3.1.2.3

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

$n = {(2 (n - 3))}^{\frac{2}{2}}$

Step 2.3.1.2.4

Cancel the common factor of $2$ .

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Step 2.3.1.2.4.1

Cancel the common factor.

$n = {(2 (n - 3))}^{\frac{2}{2}}$

Step 2.3.1.2.4.2

Rewrite the expression.

$n = {(2 (n - 3))}^{1}$

Step 2.3.1.2.5

Simplify.

$n = 2 (n - 3)$

Step 2.3.1.3

Apply the distributive property.

$n = 2 n + 2 \cdot - 3$

Step 2.3.1.4

Multiply $2$ by $- 3$ .

$n = 2 n - 6$

Step 3

Solve for $n$ .

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Step 3.1

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

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Step 3.1.1

Subtract $2 n$ from both sides of the equation.

$n - 2 n = - 6$

Step 3.1.2

Subtract $2 n$ from $n$ .

$- n = - 6$

Step 3.2

Divide each term in $- n = - 6$ by $- 1$ and simplify.

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Step 3.2.1

Divide each term in $- n = - 6$ by $- 1$ .

$\frac{- n}{- 1} = \frac{- 6}{- 1}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Dividing two negative values results in a positive value.

$\frac{n}{1} = \frac{- 6}{- 1}$

Step 3.2.2.2

Divide $n$ by $1$ .

$n = \frac{- 6}{- 1}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide $- 6$ by $- 1$ .

$n = 6$