Solve for x square root of x+6 = square root of 7x

$\sqrt{x + 6} = \sqrt{7 x}$

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

${\sqrt{x + 6}}^{2} = {\sqrt{7 x}}^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 6}$ as ${(x + 6)}^{\frac{1}{2}}$ .

${({(x + 6)}^{\frac{1}{2}})}^{2} = {\sqrt{7 x}}^{2}$

Simplify the left side.

Simplify ${({(x + 6)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(x + 6)}^{\frac{1}{2}})}^{2}$ .

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

${(x + 6)}^{\frac{1}{2} \cdot 2} = {\sqrt{7 x}}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(x + 6)}^{\frac{1}{2} \cdot 2} = {\sqrt{7 x}}^{2}$

Rewrite the expression.

${(x + 6)}^{1} = {\sqrt{7 x}}^{2}$

Simplify.

$x + 6 = {\sqrt{7 x}}^{2}$

Simplify the right side.

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

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

$x + 6 = {({(7 x)}^{\frac{1}{2}})}^{2}$

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

$x + 6 = {(7 x)}^{\frac{1}{2} \cdot 2}$

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

$x + 6 = {(7 x)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x + 6 = {(7 x)}^{\frac{2}{2}}$

Rewrite the expression.

$x + 6 = {(7 x)}^{1}$

Simplify.

$x + 6 = 7 x$

Solve for $x$ .

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

Subtract $7 x$ from both sides of the equation.

$x + 6 - 7 x = 0$

Subtract $7 x$ from $x$ .

$- 6 x + 6 = 0$

Subtract $6$ from both sides of the equation.

$- 6 x = - 6$

Divide each term in $- 6 x = - 6$ by $- 6$ and simplify.

Divide each term in $- 6 x = - 6$ by $- 6$ .

$\frac{- 6 x}{- 6} = \frac{- 6}{- 6}$

Simplify the left side.

Cancel the common factor of $- 6$ .

Cancel the common factor.

$\frac{- 6 x}{- 6} = \frac{- 6}{- 6}$

Divide $x$ by $1$ .

$x = \frac{- 6}{- 6}$

Simplify the right side.

Divide $- 6$ by $- 6$ .

$x = 1$

Solve for x square root of x+6 = square root of 7x

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