Solve for k square root of 2-k = square root of -2-3k

$\sqrt{2 - k} = \sqrt{- 2 - 3 k}$

Step 1

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

${\sqrt{2 - k}}^{2} = {\sqrt{- 2 - 3 k}}^{2}$

Step 2

Simplify each side of the equation.

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

${({(2 - k)}^{\frac{1}{2}})}^{2} = {\sqrt{- 2 - 3 k}}^{2}$

Simplify the left side.

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

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

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

${(2 - k)}^{\frac{1}{2} \cdot 2} = {\sqrt{- 2 - 3 k}}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(2 - k)}^{\frac{1}{2} \cdot 2} = {\sqrt{- 2 - 3 k}}^{2}$

Rewrite the expression.

${(2 - k)}^{1} = {\sqrt{- 2 - 3 k}}^{2}$

Simplify.

$2 - k = {\sqrt{- 2 - 3 k}}^{2}$

Simplify the right side.

Rewrite ${\sqrt{- 2 - 3 k}}^{2}$ as $- 2 - 3 k$ .

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

$2 - k = {({(- 2 - 3 k)}^{\frac{1}{2}})}^{2}$

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

$2 - k = {(- 2 - 3 k)}^{\frac{1}{2} \cdot 2}$

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

$2 - k = {(- 2 - 3 k)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$2 - k = {(- 2 - 3 k)}^{\frac{2}{2}}$

Rewrite the expression.

$2 - k = {(- 2 - 3 k)}^{1}$

Simplify.

$2 - k = - 2 - 3 k$

Step 3

Solve for $k$ .

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

Add $3 k$ to both sides of the equation.

$2 - k + 3 k = - 2$

Add $- k$ and $3 k$ .

$2 + 2 k = - 2$

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

Subtract $2$ from both sides of the equation.

$2 k = - 2 - 2$

Subtract $2$ from $- 2$ .

$2 k = - 4$

Divide each term in $2 k = - 4$ by $2$ and simplify.

Divide each term in $2 k = - 4$ by $2$ .

$\frac{2 k}{2} = \frac{- 4}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 k}{2} = \frac{- 4}{2}$

Divide $k$ by $1$ .

$k = \frac{- 4}{2}$

Simplify the right side.

Divide $- 4$ by $2$ .

$k = - 2$

Solve for k square root of 2-k = square root of -2-3k

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