Solve for g square root of 3g+1 = square root of 7g-19

$\sqrt{3 g + 1} = \sqrt{7 g - 19}$

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

${\sqrt{3 g + 1}}^{2} = {\sqrt{7 g - 19}}^{2}$

Simplify each side of the equation.

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

${({(3 g + 1)}^{\frac{1}{2}})}^{2} = {\sqrt{7 g - 19}}^{2}$

Simplify the left side.

Simplify ${({(3 g + 1)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(3 g + 1)}^{\frac{1}{2}})}^{2}$ .

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

${(3 g + 1)}^{\frac{1}{2} \cdot 2} = {\sqrt{7 g - 19}}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(3 g + 1)}^{\frac{1}{2} \cdot 2} = {\sqrt{7 g - 19}}^{2}$

Rewrite the expression.

${(3 g + 1)}^{1} = {\sqrt{7 g - 19}}^{2}$

Simplify.

$3 g + 1 = {\sqrt{7 g - 19}}^{2}$

Simplify the right side.

Rewrite ${\sqrt{7 g - 19}}^{2}$ as $7 g - 19$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7 g - 19}$ as ${(7 g - 19)}^{\frac{1}{2}}$ .

$3 g + 1 = {({(7 g - 19)}^{\frac{1}{2}})}^{2}$

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

$3 g + 1 = {(7 g - 19)}^{\frac{1}{2} \cdot 2}$

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

$3 g + 1 = {(7 g - 19)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$3 g + 1 = {(7 g - 19)}^{\frac{2}{2}}$

Rewrite the expression.

$3 g + 1 = {(7 g - 19)}^{1}$

Simplify.

$3 g + 1 = 7 g - 19$

Solve for $g$ .

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

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

$3 g + 1 - 7 g = - 19$

Subtract $7 g$ from $3 g$ .

$- 4 g + 1 = - 19$

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

Subtract $1$ from both sides of the equation.

$- 4 g = - 19 - 1$

Subtract $1$ from $- 19$ .

$- 4 g = - 20$

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

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

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

Simplify the left side.

Cancel the common factor of $- 4$ .

Cancel the common factor.

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

Divide $g$ by $1$ .

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

Simplify the right side.

Divide $- 20$ by $- 4$ .

$g = 5$

Solve for g square root of 3g+1 = square root of 7g-19

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