Algebra Examples

$\sqrt{3 - \frac{2}{3} w} + 8 = 11$

Step 1

Move all terms not containing $\sqrt{3 - \frac{2}{3} w}$ to the right side of the equation.

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Step 1.1

Subtract $8$ from both sides of the equation.

$\sqrt{3 - \frac{2}{3} w} = 11 - 8$

Step 1.2

Subtract $8$ from $11$ .

$\sqrt{3 - \frac{2}{3} w} = 3$

Step 2

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

${\sqrt{3 - \frac{2}{3} w}}^{2} = 3^{2}$

Step 3

Simplify each side of the equation.

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Step 3.1

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

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

Step 3.2

Simplify the left side.

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Step 3.2.1

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

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Step 3.2.1.1

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

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Step 3.2.1.1.1

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

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

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

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

Step 3.2.1.1.2.2

Rewrite the expression.

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

Step 3.2.1.2

Simplify each term.

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Step 3.2.1.2.1

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

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

Step 3.2.1.2.2

Move $2$ to the left of $w$ .

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

Step 3.2.1.3

Simplify.

$3 - \frac{2 w}{3} = 3^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Raise $3$ to the power of $2$ .

$3 - \frac{2 w}{3} = 9$

Step 4

Solve for $w$ .

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Step 4.1

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

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Step 4.1.1

Subtract $3$ from both sides of the equation.

$- \frac{2 w}{3} = 9 - 3$

Step 4.1.2

Subtract $3$ from $9$ .

$- \frac{2 w}{3} = 6$

Step 4.2

Multiply both sides of the equation by $- \frac{3}{2}$ .

$- \frac{3}{2} (- \frac{2 w}{3}) = - \frac{3}{2} \cdot 6$

Step 4.3

Simplify both sides of the equation.

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Step 4.3.1

Simplify the left side.

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Step 4.3.1.1

Simplify $- \frac{3}{2} (- \frac{2 w}{3})$ .

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Step 4.3.1.1.1

Cancel the common factor of $3$ .

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Step 4.3.1.1.1.1

Move the leading negative in $- \frac{3}{2}$ into the numerator.

$\frac{- 3}{2} (- \frac{2 w}{3}) = - \frac{3}{2} \cdot 6$

Step 4.3.1.1.1.2

Move the leading negative in $- \frac{2 w}{3}$ into the numerator.

$\frac{- 3}{2} \cdot \frac{- 2 w}{3} = - \frac{3}{2} \cdot 6$

Step 4.3.1.1.1.3

Factor $3$ out of $- 3$ .

$\frac{3 (- 1)}{2} \cdot \frac{- 2 w}{3} = - \frac{3}{2} \cdot 6$

Step 4.3.1.1.1.4

Cancel the common factor.

$\frac{3 \cdot - 1}{2} \cdot \frac{- 2 w}{3} = - \frac{3}{2} \cdot 6$

Step 4.3.1.1.1.5

Rewrite the expression.

$\frac{- 1}{2} (- 2 w) = - \frac{3}{2} \cdot 6$

Step 4.3.1.1.2

Cancel the common factor of $2$ .

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Step 4.3.1.1.2.1

Factor $2$ out of $- 2 w$ .

$\frac{- 1}{2} (2 (- w)) = - \frac{3}{2} \cdot 6$

Step 4.3.1.1.2.2

Cancel the common factor.

$\frac{- 1}{2} (2 (- w)) = - \frac{3}{2} \cdot 6$

Step 4.3.1.1.2.3

Rewrite the expression.

$- - w = - \frac{3}{2} \cdot 6$

Step 4.3.1.1.3

Multiply.

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Step 4.3.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 w = - \frac{3}{2} \cdot 6$

Step 4.3.1.1.3.2

Multiply $w$ by $1$ .

$w = - \frac{3}{2} \cdot 6$

Step 4.3.2

Simplify the right side.

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Step 4.3.2.1

Simplify $- \frac{3}{2} \cdot 6$ .

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Step 4.3.2.1.1

Cancel the common factor of $2$ .

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Step 4.3.2.1.1.1

Move the leading negative in $- \frac{3}{2}$ into the numerator.

$w = \frac{- 3}{2} \cdot 6$

Step 4.3.2.1.1.2

Factor $2$ out of $6$ .

$w = \frac{- 3}{2} \cdot (2 (3))$

Step 4.3.2.1.1.3

Cancel the common factor.

$w = \frac{- 3}{2} \cdot (2 \cdot 3)$

Step 4.3.2.1.1.4

Rewrite the expression.

$w = - 3 \cdot 3$

Step 4.3.2.1.2

Multiply $- 3$ by $3$ .

$w = - 9$