Algebra Examples

$\sqrt{\frac{3}{2} z + 10} = c z + 8$

Step 1

Rewrite the equation as $c z + 8 = \sqrt{\frac{3}{2} z + 10}$ .

$c z + 8 = \sqrt{\frac{3}{2} z + 10}$

Step 2

Simplify $\sqrt{\frac{3}{2} z + 10}$ .

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Step 2.1

Rewrite.

$c z + 8 = 0 + 0 + \sqrt{\frac{3}{2} z + 10}$

Step 2.2

Simplify by adding zeros.

$c z + 8 = \sqrt{\frac{3}{2} z + 10}$

Step 2.3

Combine $\frac{3}{2}$ and $z$ .

$c z + 8 = \sqrt{\frac{3 z}{2} + 10}$

Step 2.4

To write $10$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$c z + 8 = \sqrt{\frac{3 z}{2} + 10 \cdot \frac{2}{2}}$

Step 2.5

Combine $10$ and $\frac{2}{2}$ .

$c z + 8 = \sqrt{\frac{3 z}{2} + \frac{10 \cdot 2}{2}}$

Step 2.6

Combine the numerators over the common denominator.

$c z + 8 = \sqrt{\frac{3 z + 10 \cdot 2}{2}}$

Step 2.7

Multiply $10$ by $2$ .

$c z + 8 = \sqrt{\frac{3 z + 20}{2}}$

Step 2.8

Rewrite $\sqrt{\frac{3 z + 20}{2}}$ as $\frac{\sqrt{3 z + 20}}{\sqrt{2}}$ .

$c z + 8 = \frac{\sqrt{3 z + 20}}{\sqrt{2}}$

Step 2.9

Multiply $\frac{\sqrt{3 z + 20}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$c z + 8 = \frac{\sqrt{3 z + 20}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 2.10

Combine and simplify the denominator.

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Step 2.10.1

Multiply $\frac{\sqrt{3 z + 20}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$c z + 8 = \frac{\sqrt{3 z + 20} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 2.10.2

Raise $\sqrt{2}$ to the power of $1$ .

$c z + 8 = \frac{\sqrt{3 z + 20} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 2.10.3

Raise $\sqrt{2}$ to the power of $1$ .

$c z + 8 = \frac{\sqrt{3 z + 20} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 2.10.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$c z + 8 = \frac{\sqrt{3 z + 20} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 2.10.5

Add $1$ and $1$ .

$c z + 8 = \frac{\sqrt{3 z + 20} \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 2.10.6

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

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Step 2.10.6.1

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

$c z + 8 = \frac{\sqrt{3 z + 20} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 2.10.6.2

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

$c z + 8 = \frac{\sqrt{3 z + 20} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 2.10.6.3

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

$c z + 8 = \frac{\sqrt{3 z + 20} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 2.10.6.4

Cancel the common factor of $2$ .

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Step 2.10.6.4.1

Cancel the common factor.

$c z + 8 = \frac{\sqrt{3 z + 20} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 2.10.6.4.2

Rewrite the expression.

$c z + 8 = \frac{\sqrt{3 z + 20} \sqrt{2}}{2^{1}}$

Step 2.10.6.5

Evaluate the exponent.

$c z + 8 = \frac{\sqrt{3 z + 20} \sqrt{2}}{2}$

Step 2.11

Combine using the product rule for radicals.

$c z + 8 = \frac{\sqrt{(3 z + 20) \cdot 2}}{2}$

Step 2.12

Reorder factors in $\frac{\sqrt{(3 z + 20) \cdot 2}}{2}$ .

$c z + 8 = \frac{\sqrt{2 (3 z + 20)}}{2}$

Step 3

Subtract $8$ from both sides of the equation.

$c z = \frac{\sqrt{2 (3 z + 20)}}{2} - 8$

Step 4

Divide each term in $c z = \frac{\sqrt{2 (3 z + 20)}}{2} - 8$ by $z$ and simplify.

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Step 4.1

Divide each term in $c z = \frac{\sqrt{2 (3 z + 20)}}{2} - 8$ by $z$ .

$\frac{c z}{z} = \frac{\frac{\sqrt{2 (3 z + 20)}}{2}}{z} + \frac{- 8}{z}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $z$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{c z}{z} = \frac{\frac{\sqrt{2 (3 z + 20)}}{2}}{z} + \frac{- 8}{z}$

Step 4.2.1.2

Divide $c$ by $1$ .

$c = \frac{\frac{\sqrt{2 (3 z + 20)}}{2}}{z} + \frac{- 8}{z}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Combine the numerators over the common denominator.

$c = \frac{\frac{\sqrt{2 (3 z + 20)}}{2} - 8}{z}$

Step 4.3.2

To write $- 8$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$c = \frac{\frac{\sqrt{2 (3 z + 20)}}{2} - 8 \cdot \frac{2}{2}}{z}$

Step 4.3.3

Combine $- 8$ and $\frac{2}{2}$ .

$c = \frac{\frac{\sqrt{2 (3 z + 20)}}{2} + \frac{- 8 \cdot 2}{2}}{z}$

Step 4.3.4

Combine the numerators over the common denominator.

$c = \frac{\frac{\sqrt{2 (3 z + 20)} - 8 \cdot 2}{2}}{z}$

Step 4.3.5

Multiply $- 8$ by $2$ .

$c = \frac{\frac{\sqrt{2 (3 z + 20)} - 16}{2}}{z}$

Step 4.3.6

Multiply the numerator by the reciprocal of the denominator.

$c = \frac{\sqrt{2 (3 z + 20)} - 16}{2} \cdot \frac{1}{z}$

Step 4.3.7

Multiply $\frac{\sqrt{2 (3 z + 20)} - 16}{2}$ by $\frac{1}{z}$ .

$c = \frac{\sqrt{2 (3 z + 20)} - 16}{2 z}$