Solve the Rational Equation for v 11=2+ square root of 90-v

$11 = 2 + \sqrt{90 - v}$

Rewrite the equation as $2 + \sqrt{90 - v} = 11$ .

$2 + \sqrt{90 - v} = 11$

Move all terms not containing $\sqrt{90 - v}$ to the right side of the equation.

Subtract $2$ from both sides of the equation.

$\sqrt{90 - v} = 11 - 2$

Subtract $2$ from $11$ .

$\sqrt{90 - v} = 9$

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

${\sqrt{90 - v}}^{2} = 9^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{90 - v}$ as ${(90 - v)}^{\frac{1}{2}}$ .

${({(90 - v)}^{\frac{1}{2}})}^{2} = 9^{2}$

Simplify the left side.

Simplify ${({(90 - v)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(90 - v)}^{\frac{1}{2}})}^{2}$ .

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

${(90 - v)}^{\frac{1}{2} \cdot 2} = 9^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(90 - v)}^{\frac{1}{2} \cdot 2} = 9^{2}$

Rewrite the expression.

${(90 - v)}^{1} = 9^{2}$

Simplify.

$90 - v = 9^{2}$

Simplify the right side.

Raise $9$ to the power of $2$ .

$90 - v = 81$

Solve for $v$ .

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

Subtract $90$ from both sides of the equation.

$- v = 81 - 90$

Subtract $90$ from $81$ .

$- v = - 9$

Divide each term in $- v = - 9$ by $- 1$ and simplify.

Divide each term in $- v = - 9$ by $- 1$ .

$\frac{- v}{- 1} = \frac{- 9}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{v}{1} = \frac{- 9}{- 1}$

Divide $v$ by $1$ .

$v = \frac{- 9}{- 1}$

Simplify the right side.

Divide $- 9$ by $- 1$ .

$v = 9$