Solve for y square root of y+1 = square root of 4y-8

$\sqrt{y + 1} = \sqrt{4 y - 8}$

Step 1

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

${\sqrt{y + 1}}^{2} = {\sqrt{4 y - 8}}^{2}$

Step 2

Simplify each side of the equation.

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

${({(y + 1)}^{\frac{1}{2}})}^{2} = {\sqrt{4 y - 8}}^{2}$

Simplify the left side.

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

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

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

${(y + 1)}^{\frac{1}{2} \cdot 2} = {\sqrt{4 y - 8}}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(y + 1)}^{\frac{1}{2} \cdot 2} = {\sqrt{4 y - 8}}^{2}$

Rewrite the expression.

${(y + 1)}^{1} = {\sqrt{4 y - 8}}^{2}$

Simplify.

$y + 1 = {\sqrt{4 y - 8}}^{2}$

Simplify the right side.

Simplify ${\sqrt{4 y - 8}}^{2}$ .

Factor $4$ out of $4 y - 8$ .

Factor $4$ out of $4 y$ .

$y + 1 = {\sqrt{4 (y) - 8}}^{2}$

Factor $4$ out of $- 8$ .

$y + 1 = {\sqrt{4 y + 4 \cdot - 2}}^{2}$

Factor $4$ out of $4 y + 4 \cdot - 2$ .

$y + 1 = {\sqrt{4 (y - 2)}}^{2}$

Rewrite $4$ as $2^{2}$ .

$y + 1 = {\sqrt{2^{2} (y - 2)}}^{2}$

Pull terms out from under the radical.

$y + 1 = {(2 \sqrt{y - 2})}^{2}$

Simplify the expression.

Apply the product rule to $2 \sqrt{y - 2}$ .

$y + 1 = 2^{2} {\sqrt{y - 2}}^{2}$

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

$y + 1 = 4 {\sqrt{y - 2}}^{2}$

Rewrite ${\sqrt{y - 2}}^{2}$ as $y - 2$ .

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

$y + 1 = 4 {({(y - 2)}^{\frac{1}{2}})}^{2}$

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

$y + 1 = 4 {(y - 2)}^{\frac{1}{2} \cdot 2}$

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

$y + 1 = 4 {(y - 2)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$y + 1 = 4 {(y - 2)}^{\frac{2}{2}}$

Rewrite the expression.

$y + 1 = 4 {(y - 2)}^{1}$

Simplify.

$y + 1 = 4 (y - 2)$

Apply the distributive property.

$y + 1 = 4 y + 4 \cdot - 2$

Multiply $4$ by $- 2$ .

$y + 1 = 4 y - 8$

Step 3

Solve for $y$ .

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

Subtract $4 y$ from both sides of the equation.

$y + 1 - 4 y = - 8$

Subtract $4 y$ from $y$ .

$- 3 y + 1 = - 8$

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

Subtract $1$ from both sides of the equation.

$- 3 y = - 8 - 1$

Subtract $1$ from $- 8$ .

$- 3 y = - 9$

Divide each term in $- 3 y = - 9$ by $- 3$ and simplify.

Divide each term in $- 3 y = - 9$ by $- 3$ .

$\frac{- 3 y}{- 3} = \frac{- 9}{- 3}$

Simplify the left side.

Cancel the common factor of $- 3$ .

Cancel the common factor.

$\frac{- 3 y}{- 3} = \frac{- 9}{- 3}$

Divide $y$ by $1$ .

$y = \frac{- 9}{- 3}$

Simplify the right side.

Divide $- 9$ by $- 3$ .

$y = 3$

Solve for y square root of y+1 = square root of 4y-8

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