Basic Math Examples

$\sqrt{3 z + 2} = \sqrt{8 z - 8}$

Step 1

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

${\sqrt{3 z + 2}}^{2} = {\sqrt{8 z - 8}}^{2}$

Step 2

Simplify each side of the equation.

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Step 2.1

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

${({(3 z + 2)}^{\frac{1}{2}})}^{2} = {\sqrt{8 z - 8}}^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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Step 2.2.1.1

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

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Step 2.2.1.1.1

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

${(3 z + 2)}^{\frac{1}{2} \cdot 2} = {\sqrt{8 z - 8}}^{2}$

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

${(3 z + 2)}^{\frac{1}{2} \cdot 2} = {\sqrt{8 z - 8}}^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(3 z + 2)}^{1} = {\sqrt{8 z - 8}}^{2}$

Step 2.2.1.2

Simplify.

$3 z + 2 = {\sqrt{8 z - 8}}^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify ${\sqrt{8 z - 8}}^{2}$ .

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Step 2.3.1.1

Factor $8$ out of $8 z - 8$ .

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Step 2.3.1.1.1

Factor $8$ out of $8 z$ .

$3 z + 2 = {\sqrt{8 (z) - 8}}^{2}$

Step 2.3.1.1.2

Factor $8$ out of $- 8$ .

$3 z + 2 = {\sqrt{8 (z) + 8 (- 1)}}^{2}$

Step 2.3.1.1.3

Factor $8$ out of $8 (z) + 8 (- 1)$ .

$3 z + 2 = {\sqrt{8 (z - 1)}}^{2}$

Step 2.3.1.2

Rewrite $8 (z - 1)$ as $2^{2} \cdot (2 (z - 1))$ .

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Step 2.3.1.2.1

Factor $4$ out of $8$ .

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

Step 2.3.1.2.2

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

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

Step 2.3.1.2.3

Add parentheses.

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

Step 2.3.1.3

Pull terms out from under the radical.

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

Step 2.3.1.4

Simplify the expression.

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Step 2.3.1.4.1

Apply the product rule to $2 \sqrt{2 (z - 1)}$ .

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

Step 2.3.1.4.2

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

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

Step 2.3.1.5

Rewrite ${\sqrt{2 (z - 1)}}^{2}$ as $2 (z - 1)$ .

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Step 2.3.1.5.1

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

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

Step 2.3.1.5.2

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

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

Step 2.3.1.5.3

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

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

Step 2.3.1.5.4

Cancel the common factor of $2$ .

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Step 2.3.1.5.4.1

Cancel the common factor.

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

Step 2.3.1.5.4.2

Rewrite the expression.

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

Step 2.3.1.5.5

Simplify.

$3 z + 2 = 4 (2 (z - 1))$

Step 2.3.1.6

Apply the distributive property.

$3 z + 2 = 4 (2 z + 2 \cdot - 1)$

Step 2.3.1.7

Multiply $2$ by $- 1$ .

$3 z + 2 = 4 (2 z - 2)$

Step 2.3.1.8

Apply the distributive property.

$3 z + 2 = 4 (2 z) + 4 \cdot - 2$

Step 2.3.1.9

Multiply.

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Step 2.3.1.9.1

Multiply $2$ by $4$ .

$3 z + 2 = 8 z + 4 \cdot - 2$

Step 2.3.1.9.2

Multiply $4$ by $- 2$ .

$3 z + 2 = 8 z - 8$

Step 3

Solve for $z$ .

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Step 3.1

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

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Step 3.1.1

Subtract $8 z$ from both sides of the equation.

$3 z + 2 - 8 z = - 8$

Step 3.1.2

Subtract $8 z$ from $3 z$ .

$- 5 z + 2 = - 8$

Step 3.2

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

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Step 3.2.1

Subtract $2$ from both sides of the equation.

$- 5 z = - 8 - 2$

Step 3.2.2

Subtract $2$ from $- 8$ .

$- 5 z = - 10$

Step 3.3

Divide each term in $- 5 z = - 10$ by $- 5$ and simplify.

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Step 3.3.1

Divide each term in $- 5 z = - 10$ by $- 5$ .

$\frac{- 5 z}{- 5} = \frac{- 10}{- 5}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $- 5$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{- 5 z}{- 5} = \frac{- 10}{- 5}$

Step 3.3.2.1.2

Divide $z$ by $1$ .

$z = \frac{- 10}{- 5}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Divide $- 10$ by $- 5$ .

$z = 2$