Pre-Algebra Examples

$\sqrt{\frac{a}{3} - 3} = - 32$

Step 1

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

${\sqrt{\frac{a}{3} - 3}}^{2} = {(- 32)}^{2}$

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 $\sqrt{\frac{a}{3} - 3}$ as ${(\frac{a}{3} - 3)}^{\frac{1}{2}}$ .

${({(\frac{a}{3} - 3)}^{\frac{1}{2}})}^{2} = {(- 32)}^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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Step 2.2.1.1

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

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Step 2.2.1.1.1

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

${(\frac{a}{3} - 3)}^{\frac{1}{2} \cdot 2} = {(- 32)}^{2}$

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.

${(\frac{a}{3} - 3)}^{\frac{1}{2} \cdot 2} = {(- 32)}^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(\frac{a}{3} - 3)}^{1} = {(- 32)}^{2}$

Step 2.2.1.2

Simplify.

$\frac{a}{3} - 3 = {(- 32)}^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Raise $- 32$ to the power of $2$ .

$\frac{a}{3} - 3 = 1024$

Step 3

Solve for $a$ .

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Step 3.1

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

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Step 3.1.1

Add $3$ to both sides of the equation.

$\frac{a}{3} = 1024 + 3$

Step 3.1.2

Add $1024$ and $3$ .

$\frac{a}{3} = 1027$

Step 3.2

Multiply both sides of the equation by $3$ .

$3 \frac{a}{3} = 3 \cdot 1027$

Step 3.3

Simplify both sides of the equation.

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Step 3.3.1

Simplify the left side.

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Step 3.3.1.1

Cancel the common factor of $3$ .

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Step 3.3.1.1.1

Cancel the common factor.

$3 \frac{a}{3} = 3 \cdot 1027$

Step 3.3.1.1.2

Rewrite the expression.

$a = 3 \cdot 1027$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Multiply $3$ by $1027$ .

$a = 3081$

Step 4

Exclude the solutions that do not make $\sqrt{\frac{a}{3} - 3} = - 32$ true.

$No solution$