Solve the Rational Equation for x square root of 11-x = square root of x-7

$\sqrt{11 - x} = \sqrt{x - 7}$

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

${\sqrt{11 - x}}^{2} = {\sqrt{x - 7}}^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11 - x}$ as ${(11 - x)}^{\frac{1}{2}}$ .

${({(11 - x)}^{\frac{1}{2}})}^{2} = {\sqrt{x - 7}}^{2}$

Simplify the left side.

Simplify ${({(11 - x)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(11 - x)}^{\frac{1}{2}})}^{2}$ .

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

${(11 - x)}^{\frac{1}{2} \cdot 2} = {\sqrt{x - 7}}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(11 - x)}^{\frac{1}{2} \cdot 2} = {\sqrt{x - 7}}^{2}$

Rewrite the expression.

${(11 - x)}^{1} = {\sqrt{x - 7}}^{2}$

Simplify.

$11 - x = {\sqrt{x - 7}}^{2}$

Simplify the right side.

Rewrite ${\sqrt{x - 7}}^{2}$ as $x - 7$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 7}$ as ${(x - 7)}^{\frac{1}{2}}$ .

$11 - x = {({(x - 7)}^{\frac{1}{2}})}^{2}$

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

$11 - x = {(x - 7)}^{\frac{1}{2} \cdot 2}$

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

$11 - x = {(x - 7)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$11 - x = {(x - 7)}^{\frac{2}{2}}$

Rewrite the expression.

$11 - x = {(x - 7)}^{1}$

Simplify.

$11 - x = x - 7$

Solve for $x$ .

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

Subtract $x$ from both sides of the equation.

$11 - x - x = - 7$

Subtract $x$ from $- x$ .

$11 - 2 x = - 7$

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

Subtract $11$ from both sides of the equation.

$- 2 x = - 7 - 11$

Subtract $11$ from $- 7$ .

$- 2 x = - 18$

Divide each term in $- 2 x = - 18$ by $- 2$ and simplify.

Divide each term in $- 2 x = - 18$ by $- 2$ .

$\frac{- 2 x}{- 2} = \frac{- 18}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 x}{- 2} = \frac{- 18}{- 2}$

Divide $x$ by $1$ .

$x = \frac{- 18}{- 2}$

Simplify the right side.

Divide $- 18$ by $- 2$ .

$x = 9$

Solve the Rational Equation for x square root of 11-x = square root of x-7

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