Solve for n square root of 4n = square root of 3n+9

$\sqrt{4 n} = \sqrt{3 n + 9}$

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

${\sqrt{4 n}}^{2} = {\sqrt{3 n + 9}}^{2}$

Simplify each side of the equation.

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

${({(4 n)}^{\frac{1}{2}})}^{2} = {\sqrt{3 n + 9}}^{2}$

Simplify the left side.

Simplify ${({(4 n)}^{\frac{1}{2}})}^{2}$ .

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

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

${(4 n)}^{\frac{1}{2} \cdot 2} = {\sqrt{3 n + 9}}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(4 n)}^{\frac{1}{2} \cdot 2} = {\sqrt{3 n + 9}}^{2}$

Rewrite the expression.

${(4 n)}^{1} = {\sqrt{3 n + 9}}^{2}$

Simplify.

$4 n = {\sqrt{3 n + 9}}^{2}$

Simplify the right side.

Simplify ${\sqrt{3 n + 9}}^{2}$ .

Factor $3$ out of $3 n + 9$ .

Factor $3$ out of $3 n$ .

$4 n = {\sqrt{3 (n) + 9}}^{2}$

Factor $3$ out of $9$ .

$4 n = {\sqrt{3 n + 3 \cdot 3}}^{2}$

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

$4 n = {\sqrt{3 (n + 3)}}^{2}$

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$4 n = {(3 (n + 3))}^{1}$

Simplify.

$4 n = 3 (n + 3)$

Apply the distributive property.

$4 n = 3 n + 3 \cdot 3$

Multiply $3$ by $3$ .

$4 n = 3 n + 9$

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

Subtract $3 n$ from both sides of the equation.

$4 n - 3 n = 9$

Subtract $3 n$ from $4 n$ .

$n = 9$

Solve for n square root of 4n = square root of 3n+9

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