Solve for n square root of 15-n = square root of 3n-5

$\sqrt{15 - n} = \sqrt{3 n - 5}$

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

${\sqrt{15 - n}}^{2} = {\sqrt{3 n - 5}}^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{15 - n}$ as ${(15 - n)}^{\frac{1}{2}}$ .

${({(15 - n)}^{\frac{1}{2}})}^{2} = {\sqrt{3 n - 5}}^{2}$

Simplify the left side.

Simplify ${({(15 - n)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(15 - n)}^{\frac{1}{2}})}^{2}$ .

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

${(15 - n)}^{\frac{1}{2} \cdot 2} = {\sqrt{3 n - 5}}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(15 - n)}^{\frac{1}{2} \cdot 2} = {\sqrt{3 n - 5}}^{2}$

Rewrite the expression.

${(15 - n)}^{1} = {\sqrt{3 n - 5}}^{2}$

Simplify.

$15 - n = {\sqrt{3 n - 5}}^{2}$

Simplify the right side.

Rewrite ${\sqrt{3 n - 5}}^{2}$ as $3 n - 5$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 n - 5}$ as ${(3 n - 5)}^{\frac{1}{2}}$ .

$15 - n = {({(3 n - 5)}^{\frac{1}{2}})}^{2}$

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

$15 - n = {(3 n - 5)}^{\frac{1}{2} \cdot 2}$

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

$15 - n = {(3 n - 5)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$15 - n = {(3 n - 5)}^{\frac{2}{2}}$

Rewrite the expression.

$15 - n = {(3 n - 5)}^{1}$

Simplify.

$15 - n = 3 n - 5$

Solve for $n$ .

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

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

$15 - n - 3 n = - 5$

Subtract $3 n$ from $- n$ .

$15 - 4 n = - 5$

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

Subtract $15$ from both sides of the equation.

$- 4 n = - 5 - 15$

Subtract $15$ from $- 5$ .

$- 4 n = - 20$

Divide each term in $- 4 n = - 20$ by $- 4$ and simplify.

Divide each term in $- 4 n = - 20$ by $- 4$ .

$\frac{- 4 n}{- 4} = \frac{- 20}{- 4}$

Simplify the left side.

Cancel the common factor of $- 4$ .

Cancel the common factor.

$\frac{- 4 n}{- 4} = \frac{- 20}{- 4}$

Divide $n$ by $1$ .

$n = \frac{- 20}{- 4}$

Simplify the right side.

Divide $- 20$ by $- 4$ .

$n = 5$

Solve for n square root of 15-n = square root of 3n-5

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