Solve for z square root of 3z-2 = square root of z-4

$\sqrt{3 z - 2} = \sqrt{z - 4}$

Step 1

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

${\sqrt{3 z - 2}}^{2} = {\sqrt{z - 4}}^{2}$

Step 2

Simplify each side of the equation.

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

${({(3 z - 2)}^{\frac{1}{2}})}^{2} = {\sqrt{z - 4}}^{2}$

Simplify the left side.

Simplify ${({(3 z - 2)}^{\frac{1}{2}})}^{2}$ .

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

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

${(3 z - 2)}^{\frac{1}{2} \cdot 2} = {\sqrt{z - 4}}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(3 z - 2)}^{\frac{1}{2} \cdot 2} = {\sqrt{z - 4}}^{2}$

Rewrite the expression.

${(3 z - 2)}^{1} = {\sqrt{z - 4}}^{2}$

Simplify.

$3 z - 2 = {\sqrt{z - 4}}^{2}$

Simplify the right side.

Rewrite ${\sqrt{z - 4}}^{2}$ as $z - 4$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{z - 4}$ as ${(z - 4)}^{\frac{1}{2}}$ .

$3 z - 2 = {({(z - 4)}^{\frac{1}{2}})}^{2}$

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

$3 z - 2 = {(z - 4)}^{\frac{1}{2} \cdot 2}$

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

$3 z - 2 = {(z - 4)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$3 z - 2 = {(z - 4)}^{\frac{2}{2}}$

Rewrite the expression.

$3 z - 2 = {(z - 4)}^{1}$

Simplify.

$3 z - 2 = z - 4$

Step 3

Solve for $z$ .

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

Subtract $z$ from both sides of the equation.

$3 z - 2 - z = - 4$

Subtract $z$ from $3 z$ .

$2 z - 2 = - 4$

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

Add $2$ to both sides of the equation.

$2 z = - 4 + 2$

Add $- 4$ and $2$ .

$2 z = - 2$

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

Divide each term in $2 z = - 2$ by $2$ .

$\frac{2 z}{2} = \frac{- 2}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 z}{2} = \frac{- 2}{2}$

Divide $z$ by $1$ .

$z = \frac{- 2}{2}$

Simplify the right side.

Divide $- 2$ by $2$ .

$z = - 1$

Step 4

Exclude the solutions that do not make $\sqrt{3 z - 2} = \sqrt{z - 4}$ true.

$No solution$

Solve for z square root of 3z-2 = square root of z-4

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