Algebra Examples

$\sqrt{5 - \frac{w}{4}} = \sqrt{w + 8}$

Step 1

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

${\sqrt{5 - \frac{w}{4}}}^{2} = {\sqrt{w + 8}}^{2}$

Step 2

Simplify each side of the equation.

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Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 - \frac{w}{4}}$ as ${(5 - \frac{w}{4})}^{\frac{1}{2}}$ .

${({(5 - \frac{w}{4})}^{\frac{1}{2}})}^{2} = {\sqrt{w + 8}}^{2}$

Simplify the left side.

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Simplify ${({(5 - \frac{w}{4})}^{\frac{1}{2}})}^{2}$ .

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Multiply the exponents in ${({(5 - \frac{w}{4})}^{\frac{1}{2}})}^{2}$ .

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

${(5 - \frac{w}{4})}^{\frac{1}{2} \cdot 2} = {\sqrt{w + 8}}^{2}$

Cancel the common factor of $2$ .

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Cancel the common factor.

${(5 - \frac{w}{4})}^{\frac{1}{2} \cdot 2} = {\sqrt{w + 8}}^{2}$

Rewrite the expression.

${(5 - \frac{w}{4})}^{1} = {\sqrt{w + 8}}^{2}$

Simplify.

$5 - \frac{w}{4} = {\sqrt{w + 8}}^{2}$

Simplify the right side.

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Rewrite ${\sqrt{w + 8}}^{2}$ as $w + 8$ .

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Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{w + 8}$ as ${(w + 8)}^{\frac{1}{2}}$ .

$5 - \frac{w}{4} = {({(w + 8)}^{\frac{1}{2}})}^{2}$

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

$5 - \frac{w}{4} = {(w + 8)}^{\frac{1}{2} \cdot 2}$

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

$5 - \frac{w}{4} = {(w + 8)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$5 - \frac{w}{4} = {(w + 8)}^{\frac{2}{2}}$

Rewrite the expression.

$5 - \frac{w}{4} = {(w + 8)}^{1}$

Simplify.

$5 - \frac{w}{4} = w + 8$

Step 3

Solve for $w$ .

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

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Subtract $w$ from both sides of the equation.

$5 - \frac{w}{4} - w = 8$

To write $- w$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$5 - \frac{w}{4} - w \cdot \frac{4}{4} = 8$

Combine $- w$ and $\frac{4}{4}$ .

$5 - \frac{w}{4} + \frac{- w \cdot 4}{4} = 8$

Combine the numerators over the common denominator.

$5 + \frac{- w - w \cdot 4}{4} = 8$

Simplify each term.

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Simplify the numerator.

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Factor $w$ out of $- w - w \cdot 4$ .

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Factor $w$ out of $- w$ .

$5 + \frac{w \cdot - 1 - w \cdot 4}{4} = 8$

Factor $w$ out of $- w \cdot 4$ .

$5 + \frac{w \cdot - 1 + w (- 1 \cdot 4)}{4} = 8$

Factor $w$ out of $w \cdot - 1 + w (- 1 \cdot 4)$ .

$5 + \frac{w (- 1 - 1 \cdot 4)}{4} = 8$

Multiply $- 1$ by $4$ .

$5 + \frac{w (- 1 - 4)}{4} = 8$

Subtract $4$ from $- 1$ .

$5 + \frac{w \cdot - 5}{4} = 8$

Move $- 5$ to the left of $w$ .

$5 + \frac{- 5 \cdot w}{4} = 8$

Move the negative in front of the fraction.

$5 - \frac{5 w}{4} = 8$

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

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Subtract $5$ from both sides of the equation.

$- \frac{5 w}{4} = 8 - 5$

Subtract $5$ from $8$ .

$- \frac{5 w}{4} = 3$

Multiply both sides of the equation by $- \frac{4}{5}$ .

$- \frac{4}{5} (- \frac{5 w}{4}) = - \frac{4}{5} \cdot 3$

Simplify both sides of the equation.

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Simplify the left side.

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Simplify $- \frac{4}{5} (- \frac{5 w}{4})$ .

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Cancel the common factor of $4$ .

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Move the leading negative in $- \frac{4}{5}$ into the numerator.

$\frac{- 4}{5} (- \frac{5 w}{4}) = - \frac{4}{5} \cdot 3$

Move the leading negative in $- \frac{5 w}{4}$ into the numerator.

$\frac{- 4}{5} \cdot \frac{- 5 w}{4} = - \frac{4}{5} \cdot 3$

Factor $4$ out of $- 4$ .

$\frac{4 (- 1)}{5} \cdot \frac{- 5 w}{4} = - \frac{4}{5} \cdot 3$

Cancel the common factor.

$\frac{4 \cdot - 1}{5} \cdot \frac{- 5 w}{4} = - \frac{4}{5} \cdot 3$

Rewrite the expression.

$\frac{- 1}{5} (- 5 w) = - \frac{4}{5} \cdot 3$

Cancel the common factor of $5$ .

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Factor $5$ out of $- 5 w$ .

$\frac{- 1}{5} (5 (- w)) = - \frac{4}{5} \cdot 3$

Cancel the common factor.

$\frac{- 1}{5} (5 (- w)) = - \frac{4}{5} \cdot 3$

Rewrite the expression.

$- - w = - \frac{4}{5} \cdot 3$

Multiply.

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Multiply $- 1$ by $- 1$ .

$1 w = - \frac{4}{5} \cdot 3$

Multiply $w$ by $1$ .

$w = - \frac{4}{5} \cdot 3$

Simplify the right side.

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Simplify $- \frac{4}{5} \cdot 3$ .

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Multiply $- \frac{4}{5} \cdot 3$ .

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Multiply $3$ by $- 1$ .

$w = - 3 (\frac{4}{5})$

Combine $- 3$ and $\frac{4}{5}$ .

$w = \frac{- 3 \cdot 4}{5}$

Multiply $- 3$ by $4$ .

$w = \frac{- 12}{5}$

Move the negative in front of the fraction.

$w = - \frac{12}{5}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$w = - \frac{12}{5}$

Decimal Form:

$w = - 2.4$

Mixed Number Form:

$w = - 2 \frac{2}{5}$