Solve for v square root of v-5 = square root of 23-v

$\sqrt{v - 5} = \sqrt{23 - v}$

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

${\sqrt{v - 5}}^{2} = {\sqrt{23 - v}}^{2}$

Simplify each side of the equation.

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

${({(v - 5)}^{\frac{1}{2}})}^{2} = {\sqrt{23 - v}}^{2}$

Simplify the left side.

Simplify ${({(v - 5)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(v - 5)}^{\frac{1}{2}})}^{2}$ .

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

${(v - 5)}^{\frac{1}{2} \cdot 2} = {\sqrt{23 - v}}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(v - 5)}^{\frac{1}{2} \cdot 2} = {\sqrt{23 - v}}^{2}$

Rewrite the expression.

${(v - 5)}^{1} = {\sqrt{23 - v}}^{2}$

Simplify.

$v - 5 = {\sqrt{23 - v}}^{2}$

Simplify the right side.

Rewrite ${\sqrt{23 - v}}^{2}$ as $23 - v$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{23 - v}$ as ${(23 - v)}^{\frac{1}{2}}$ .

$v - 5 = {({(23 - v)}^{\frac{1}{2}})}^{2}$

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

$v - 5 = {(23 - v)}^{\frac{1}{2} \cdot 2}$

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

$v - 5 = {(23 - v)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$v - 5 = {(23 - v)}^{\frac{2}{2}}$

Rewrite the expression.

$v - 5 = {(23 - v)}^{1}$

Simplify.

$v - 5 = 23 - v$

Solve for $v$ .

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

Add $v$ to both sides of the equation.

$v - 5 + v = 23$

Add $v$ and $v$ .

$2 v - 5 = 23$

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

Add $5$ to both sides of the equation.

$2 v = 23 + 5$

Add $23$ and $5$ .

$2 v = 28$

Divide each term in $2 v = 28$ by $2$ and simplify.

Divide each term in $2 v = 28$ by $2$ .

$\frac{2 v}{2} = \frac{28}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 v}{2} = \frac{28}{2}$

Divide $v$ by $1$ .

$v = \frac{28}{2}$

Simplify the right side.

Divide $28$ by $2$ .

$v = 14$

Solve for v square root of v-5 = square root of 23-v

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