Solve the Rational Equation for n square root of 3n+12 = square root of n+8

$\sqrt{3 n + 12} = \sqrt{n + 8}$

Step 1

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

${\sqrt{3 n + 12}}^{2} = {\sqrt{n + 8}}^{2}$

Step 2

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 n + 12}$ as ${(3 n + 12)}^{\frac{1}{2}}$ .

${({(3 n + 12)}^{\frac{1}{2}})}^{2} = {\sqrt{n + 8}}^{2}$

Simplify the left side.

Simplify ${({(3 n + 12)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(3 n + 12)}^{\frac{1}{2}})}^{2}$ .

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

${(3 n + 12)}^{\frac{1}{2} \cdot 2} = {\sqrt{n + 8}}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(3 n + 12)}^{\frac{1}{2} \cdot 2} = {\sqrt{n + 8}}^{2}$

Rewrite the expression.

${(3 n + 12)}^{1} = {\sqrt{n + 8}}^{2}$

Simplify.

$3 n + 12 = {\sqrt{n + 8}}^{2}$

Simplify the right side.

Rewrite ${\sqrt{n + 8}}^{2}$ as $n + 8$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{n + 8}$ as ${(n + 8)}^{\frac{1}{2}}$ .

$3 n + 12 = {({(n + 8)}^{\frac{1}{2}})}^{2}$

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

$3 n + 12 = {(n + 8)}^{\frac{1}{2} \cdot 2}$

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

$3 n + 12 = {(n + 8)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$3 n + 12 = {(n + 8)}^{\frac{2}{2}}$

Rewrite the expression.

$3 n + 12 = {(n + 8)}^{1}$

Simplify.

$3 n + 12 = n + 8$

Step 3

Solve for $n$ .

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

Subtract $n$ from both sides of the equation.

$3 n + 12 - n = 8$

Subtract $n$ from $3 n$ .

$2 n + 12 = 8$

Move all terms not containing $n$ to the right side of the equation.

Subtract $12$ from both sides of the equation.

$2 n = 8 - 12$

Subtract $12$ from $8$ .

$2 n = - 4$

Divide each term in $2 n = - 4$ by $2$ and simplify.

Divide each term in $2 n = - 4$ by $2$ .

$\frac{2 n}{2} = \frac{- 4}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 n}{2} = \frac{- 4}{2}$

Divide $n$ by $1$ .

$n = \frac{- 4}{2}$

Simplify the right side.

Divide $- 4$ by $2$ .

$n = - 2$

Solve the Rational Equation for n square root of 3n+12 = square root of n+8

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