Algebra Examples

$6 \sqrt{8 a^{2}} = 12$

Step 1

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

${(6 \sqrt{8 a^{2}})}^{2} = 12^{2}$

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 $\sqrt{8 a^{2}}$ as ${(8 a^{2})}^{\frac{1}{2}}$ .

${(6 {(8 a^{2})}^{\frac{1}{2}})}^{2} = 12^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${(6 {(8 a^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Apply the product rule to $8 a^{2}$ .

${(6 (8^{\frac{1}{2}} {(a^{2})}^{\frac{1}{2}}))}^{2} = 12^{2}$

Step 2.2.1.2

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

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Step 2.2.1.2.1

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

${(6 (8^{\frac{1}{2}} a^{2 (\frac{1}{2})}))}^{2} = 12^{2}$

Step 2.2.1.2.2

Cancel the common factor of $2$ .

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Step 2.2.1.2.2.1

Cancel the common factor.

${(6 (8^{\frac{1}{2}} a^{2 (\frac{1}{2})}))}^{2} = 12^{2}$

Step 2.2.1.2.2.2

Rewrite the expression.

${(6 (8^{\frac{1}{2}} a^{1}))}^{2} = 12^{2}$

Step 2.2.1.3

Simplify.

${(6 (8^{\frac{1}{2}} a))}^{2} = 12^{2}$

Step 2.2.1.4

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.2.1.4.1

Apply the product rule to $6 \cdot 8^{\frac{1}{2}} a$ .

${(6 \cdot 8^{\frac{1}{2}})}^{2} a^{2} = 12^{2}$

Step 2.2.1.4.2

Apply the product rule to $6 \cdot 8^{\frac{1}{2}}$ .

$6^{2} \cdot {(8^{\frac{1}{2}})}^{2} a^{2} = 12^{2}$

Step 2.2.1.5

Raise $6$ to the power of $2$ .

$36 \cdot {(8^{\frac{1}{2}})}^{2} a^{2} = 12^{2}$

Step 2.2.1.6

Multiply the exponents in ${(8^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.6.1

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

$36 \cdot 8^{\frac{1}{2} \cdot 2} a^{2} = 12^{2}$

Step 2.2.1.6.2

Cancel the common factor of $2$ .

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Step 2.2.1.6.2.1

Cancel the common factor.

$36 \cdot 8^{\frac{1}{2} \cdot 2} a^{2} = 12^{2}$

Step 2.2.1.6.2.2

Rewrite the expression.

$36 \cdot 8^{1} a^{2} = 12^{2}$

Step 2.2.1.7

Evaluate the exponent.

$36 \cdot 8 a^{2} = 12^{2}$

Step 2.2.1.8

Multiply $36$ by $8$ .

$288 a^{2} = 12^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Raise $12$ to the power of $2$ .

$288 a^{2} = 144$

Step 3

Solve for $a$ .

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Step 3.1

Divide each term in $288 a^{2} = 144$ by $288$ and simplify.

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Step 3.1.1

Divide each term in $288 a^{2} = 144$ by $288$ .

$\frac{288 a^{2}}{288} = \frac{144}{288}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor of $288$ .

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Step 3.1.2.1.1

Cancel the common factor.

$\frac{288 a^{2}}{288} = \frac{144}{288}$

Step 3.1.2.1.2

Divide $a^{2}$ by $1$ .

$a^{2} = \frac{144}{288}$

Step 3.1.3

Simplify the right side.

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Step 3.1.3.1

Cancel the common factor of $144$ and $288$ .

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Step 3.1.3.1.1

Factor $144$ out of $144$ .

$a^{2} = \frac{144 (1)}{288}$

Step 3.1.3.1.2

Cancel the common factors.

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Step 3.1.3.1.2.1

Factor $144$ out of $288$ .

$a^{2} = \frac{144 \cdot 1}{144 \cdot 2}$

Step 3.1.3.1.2.2

Cancel the common factor.

$a^{2} = \frac{144 \cdot 1}{144 \cdot 2}$

Step 3.1.3.1.2.3

Rewrite the expression.

$a^{2} = \frac{1}{2}$

Step 3.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$a = \pm \sqrt{\frac{1}{2}}$

Step 3.3

Simplify $\pm \sqrt{\frac{1}{2}}$ .

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Step 3.3.1

Rewrite $\sqrt{\frac{1}{2}}$ as $\frac{\sqrt{1}}{\sqrt{2}}$ .

$a = \pm \frac{\sqrt{1}}{\sqrt{2}}$

Step 3.3.2

Any root of $1$ is $1$ .

$a = \pm \frac{1}{\sqrt{2}}$

Step 3.3.3

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$a = \pm \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 3.3.4

Combine and simplify the denominator.

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Step 3.3.4.1

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$a = \pm \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 3.3.4.2

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

$a = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 3.3.4.3

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

$a = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 3.3.4.4

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

$a = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 3.3.4.5

Add $1$ and $1$ .

$a = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{2}}$

Step 3.3.4.6

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

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Step 3.3.4.6.1

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

$a = \pm \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 3.3.4.6.2

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

$a = \pm \frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 3.3.4.6.3

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

$a = \pm \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 3.3.4.6.4

Cancel the common factor of $2$ .

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Step 3.3.4.6.4.1

Cancel the common factor.

$a = \pm \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 3.3.4.6.4.2

Rewrite the expression.

$a = \pm \frac{\sqrt{2}}{2^{1}}$

Step 3.3.4.6.5

Evaluate the exponent.

$a = \pm \frac{\sqrt{2}}{2}$

Step 3.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.4.1

First, use the positive value of the $\pm$ to find the first solution.

$a = \frac{\sqrt{2}}{2}$

Step 3.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$a = - \frac{\sqrt{2}}{2}$

Step 3.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$a = \frac{\sqrt{2}}{2}, - \frac{\sqrt{2}}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$a = \frac{\sqrt{2}}{2}, - \frac{\sqrt{2}}{2}$

Decimal Form:

$a = 0.70710678 \dots, - 0.70710678 \dots$