Basic Math Examples

$a = \sqrt{11^{2} - 6 a^{2}}$

Step 1

Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.

$\sqrt{11^{2} - 6 a^{2}} = a$

Step 2

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

${\sqrt{11^{2} - 6 a^{2}}}^{2} = a^{2}$

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 $\sqrt{11^{2} - 6 a^{2}}$ as ${(11^{2} - 6 a^{2})}^{\frac{1}{2}}$ .

${({(11^{2} - 6 a^{2})}^{\frac{1}{2}})}^{2} = a^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(11^{2} - 6 a^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(11^{2} - 6 a^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

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

${(11^{2} - 6 a^{2})}^{\frac{1}{2} \cdot 2} = a^{2}$

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.

${(11^{2} - 6 a^{2})}^{\frac{1}{2} \cdot 2} = a^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(11^{2} - 6 a^{2})}^{1} = a^{2}$

Step 3.2.1.2

Raise $11$ to the power of $2$ .

${(121 - 6 a^{2})}^{1} = a^{2}$

Step 3.2.1.3

Simplify.

$121 - 6 a^{2} = a^{2}$

Step 4

Solve for $a$ .

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Step 4.1

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

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Step 4.1.1

Subtract $a^{2}$ from both sides of the equation.

$121 - 6 a^{2} - a^{2} = 0$

Step 4.1.2

Subtract $a^{2}$ from $- 6 a^{2}$ .

$121 - 7 a^{2} = 0$

Step 4.2

Subtract $121$ from both sides of the equation.

$- 7 a^{2} = - 121$

Step 4.3

Divide each term in $- 7 a^{2} = - 121$ by $- 7$ and simplify.

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Step 4.3.1

Divide each term in $- 7 a^{2} = - 121$ by $- 7$ .

$\frac{- 7 a^{2}}{- 7} = \frac{- 121}{- 7}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $- 7$ .

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Step 4.3.2.1.1

Cancel the common factor.

$\frac{- 7 a^{2}}{- 7} = \frac{- 121}{- 7}$

Step 4.3.2.1.2

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

$a^{2} = \frac{- 121}{- 7}$

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Dividing two negative values results in a positive value.

$a^{2} = \frac{121}{7}$

Step 4.4

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

$a = \pm \sqrt{\frac{121}{7}}$

Step 4.5

Simplify $\pm \sqrt{\frac{121}{7}}$ .

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Step 4.5.1

Rewrite $\sqrt{\frac{121}{7}}$ as $\frac{\sqrt{121}}{\sqrt{7}}$ .

$a = \pm \frac{\sqrt{121}}{\sqrt{7}}$

Step 4.5.2

Simplify the numerator.

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Step 4.5.2.1

Rewrite $121$ as $11^{2}$ .

$a = \pm \frac{\sqrt{11^{2}}}{\sqrt{7}}$

Step 4.5.2.2

Pull terms out from under the radical, assuming positive real numbers.

$a = \pm \frac{11}{\sqrt{7}}$

Step 4.5.3

Multiply $\frac{11}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$a = \pm \frac{11}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}}$

Step 4.5.4

Combine and simplify the denominator.

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Step 4.5.4.1

Multiply $\frac{11}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$a = \pm \frac{11 \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 4.5.4.2

Raise $\sqrt{7}$ to the power of $1$ .

$a = \pm \frac{11 \sqrt{7}}{{\sqrt{7}}^{1} \sqrt{7}}$

Step 4.5.4.3

Raise $\sqrt{7}$ to the power of $1$ .

$a = \pm \frac{11 \sqrt{7}}{{\sqrt{7}}^{1} {\sqrt{7}}^{1}}$

Step 4.5.4.4

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

$a = \pm \frac{11 \sqrt{7}}{{\sqrt{7}}^{1 + 1}}$

Step 4.5.4.5

Add $1$ and $1$ .

$a = \pm \frac{11 \sqrt{7}}{{\sqrt{7}}^{2}}$

Step 4.5.4.6

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 4.5.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$a = \pm \frac{11 \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}}$

Step 4.5.4.6.2

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

$a = \pm \frac{11 \sqrt{7}}{7^{\frac{1}{2} \cdot 2}}$

Step 4.5.4.6.3

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

$a = \pm \frac{11 \sqrt{7}}{7^{\frac{2}{2}}}$

Step 4.5.4.6.4

Cancel the common factor of $2$ .

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Step 4.5.4.6.4.1

Cancel the common factor.

$a = \pm \frac{11 \sqrt{7}}{7^{\frac{2}{2}}}$

Step 4.5.4.6.4.2

Rewrite the expression.

$a = \pm \frac{11 \sqrt{7}}{7^{1}}$

Step 4.5.4.6.5

Evaluate the exponent.

$a = \pm \frac{11 \sqrt{7}}{7}$

Step 4.6

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

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Step 4.6.1

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

$a = \frac{11 \sqrt{7}}{7}$

Step 4.6.2

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

$a = - \frac{11 \sqrt{7}}{7}$

Step 4.6.3

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

$a = \frac{11 \sqrt{7}}{7}, - \frac{11 \sqrt{7}}{7}$

Step 5

Exclude the solutions that do not make $a = \sqrt{11^{2} - 6 a^{2}}$ true.

$a = \frac{11 \sqrt{7}}{7}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$a = \frac{11 \sqrt{7}}{7}$

Decimal Form:

$a = 4.15760920 \dots$