Algebra Examples

$\sqrt{\frac{2}{3} w + 12} = 8 + c w$

Step 1

Rewrite the equation as $8 + c w = \sqrt{\frac{2}{3} w + 12}$ .

$8 + c w = \sqrt{\frac{2}{3} w + 12}$

Step 2

Simplify $\sqrt{\frac{2}{3} w + 12}$ .

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Step 2.1

Rewrite.

$8 + c w = 0 + 0 + \sqrt{\frac{2}{3} w + 12}$

Step 2.2

Simplify by adding zeros.

$8 + c w = \sqrt{\frac{2}{3} w + 12}$

Step 2.3

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

$8 + c w = \sqrt{\frac{2 w}{3} + 12}$

Step 2.4

Factor $2$ out of $\frac{2 w}{3} + 12$ .

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Step 2.4.1

Factor $2$ out of $\frac{2 w}{3}$ .

$8 + c w = \sqrt{2 \frac{w}{3} + 12}$

Step 2.4.2

Factor $2$ out of $12$ .

$8 + c w = \sqrt{2 \frac{w}{3} + 2 \cdot 6}$

Step 2.4.3

Factor $2$ out of $2 \frac{w}{3} + 2 \cdot 6$ .

$8 + c w = \sqrt{2 (\frac{w}{3} + 6)}$

Step 2.5

To write $6$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$8 + c w = \sqrt{2 (\frac{w}{3} + 6 \cdot \frac{3}{3})}$

Step 2.6

Combine $6$ and $\frac{3}{3}$ .

$8 + c w = \sqrt{2 (\frac{w}{3} + \frac{6 \cdot 3}{3})}$

Step 2.7

Combine the numerators over the common denominator.

$8 + c w = \sqrt{2 \frac{w + 6 \cdot 3}{3}}$

Step 2.8

Multiply $6$ by $3$ .

$8 + c w = \sqrt{2 \frac{w + 18}{3}}$

Step 2.9

Combine $2$ and $\frac{w + 18}{3}$ .

$8 + c w = \sqrt{\frac{2 (w + 18)}{3}}$

Step 2.10

Rewrite $\sqrt{\frac{2 (w + 18)}{3}}$ as $\frac{\sqrt{2 (w + 18)}}{\sqrt{3}}$ .

$8 + c w = \frac{\sqrt{2 (w + 18)}}{\sqrt{3}}$

Step 2.11

Multiply $\frac{\sqrt{2 (w + 18)}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$8 + c w = \frac{\sqrt{2 (w + 18)}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 2.12

Combine and simplify the denominator.

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Step 2.12.1

Multiply $\frac{\sqrt{2 (w + 18)}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$8 + c w = \frac{\sqrt{2 (w + 18)} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 2.12.2

Raise $\sqrt{3}$ to the power of $1$ .

$8 + c w = \frac{\sqrt{2 (w + 18)} \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 2.12.3

Raise $\sqrt{3}$ to the power of $1$ .

$8 + c w = \frac{\sqrt{2 (w + 18)} \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 2.12.4

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

$8 + c w = \frac{\sqrt{2 (w + 18)} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 2.12.5

Add $1$ and $1$ .

$8 + c w = \frac{\sqrt{2 (w + 18)} \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 2.12.6

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

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Step 2.12.6.1

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

$8 + c w = \frac{\sqrt{2 (w + 18)} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 2.12.6.2

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

$8 + c w = \frac{\sqrt{2 (w + 18)} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 2.12.6.3

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

$8 + c w = \frac{\sqrt{2 (w + 18)} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 2.12.6.4

Cancel the common factor of $2$ .

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Step 2.12.6.4.1

Cancel the common factor.

$8 + c w = \frac{\sqrt{2 (w + 18)} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 2.12.6.4.2

Rewrite the expression.

$8 + c w = \frac{\sqrt{2 (w + 18)} \sqrt{3}}{3^{1}}$

Step 2.12.6.5

Evaluate the exponent.

$8 + c w = \frac{\sqrt{2 (w + 18)} \sqrt{3}}{3}$

Step 2.13

Simplify the numerator.

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Step 2.13.1

Combine using the product rule for radicals.

$8 + c w = \frac{\sqrt{2 (w + 18) \cdot 3}}{3}$

Step 2.13.2

Multiply $3$ by $2$ .

$8 + c w = \frac{\sqrt{6 (w + 18)}}{3}$

Step 3

Subtract $8$ from both sides of the equation.

$c w = \frac{\sqrt{6 (w + 18)}}{3} - 8$

Step 4

Divide each term in $c w = \frac{\sqrt{6 (w + 18)}}{3} - 8$ by $w$ and simplify.

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Step 4.1

Divide each term in $c w = \frac{\sqrt{6 (w + 18)}}{3} - 8$ by $w$ .

$\frac{c w}{w} = \frac{\frac{\sqrt{6 (w + 18)}}{3}}{w} + \frac{- 8}{w}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $w$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{c w}{w} = \frac{\frac{\sqrt{6 (w + 18)}}{3}}{w} + \frac{- 8}{w}$

Step 4.2.1.2

Divide $c$ by $1$ .

$c = \frac{\frac{\sqrt{6 (w + 18)}}{3}}{w} + \frac{- 8}{w}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Combine the numerators over the common denominator.

$c = \frac{\frac{\sqrt{6 (w + 18)}}{3} - 8}{w}$

Step 4.3.2

To write $- 8$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$c = \frac{\frac{\sqrt{6 (w + 18)}}{3} - 8 \cdot \frac{3}{3}}{w}$

Step 4.3.3

Combine $- 8$ and $\frac{3}{3}$ .

$c = \frac{\frac{\sqrt{6 (w + 18)}}{3} + \frac{- 8 \cdot 3}{3}}{w}$

Step 4.3.4

Combine the numerators over the common denominator.

$c = \frac{\frac{\sqrt{6 (w + 18)} - 8 \cdot 3}{3}}{w}$

Step 4.3.5

Multiply $- 8$ by $3$ .

$c = \frac{\frac{\sqrt{6 (w + 18)} - 24}{3}}{w}$

Step 4.3.6

Multiply the numerator by the reciprocal of the denominator.

$c = \frac{\sqrt{6 (w + 18)} - 24}{3} \cdot \frac{1}{w}$

Step 4.3.7

Multiply $\frac{\sqrt{6 (w + 18)} - 24}{3}$ by $\frac{1}{w}$ .

$c = \frac{\sqrt{6 (w + 18)} - 24}{3 w}$