Basic Math Examples

$8^{2} = a^{2} + 11 {(\sqrt{5})}^{2}$

Step 1

Rewrite the equation as $a^{2} + 11 {\sqrt{5}}^{2} = 8^{2}$ .

$a^{2} + 11 {\sqrt{5}}^{2} = 8^{2}$

Step 2

Simplify each term.

Tap for more steps...

Step 2.1

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

Tap for more steps...

Step 2.1.1

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

$a^{2} + 11 {(5^{\frac{1}{2}})}^{2} = 8^{2}$

Step 2.1.2

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

$a^{2} + 11 \cdot 5^{\frac{1}{2} \cdot 2} = 8^{2}$

Step 2.1.3

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

$a^{2} + 11 \cdot 5^{\frac{2}{2}} = 8^{2}$

Step 2.1.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.1.4.1

Cancel the common factor.

$a^{2} + 11 \cdot 5^{\frac{2}{2}} = 8^{2}$

Step 2.1.4.2

Rewrite the expression.

$a^{2} + 11 \cdot 5^{1} = 8^{2}$

Step 2.1.5

Evaluate the exponent.

$a^{2} + 11 \cdot 5 = 8^{2}$

Step 2.2

Multiply $11$ by $5$ .

$a^{2} + 55 = 8^{2}$

Step 3

Raise $8$ to the power of $2$ .

$a^{2} + 55 = 64$

Step 4

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

Tap for more steps...

Step 4.1

Subtract $55$ from both sides of the equation.

$a^{2} = 64 - 55$

Step 4.2

Subtract $55$ from $64$ .

$a^{2} = 9$

Step 5

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

$a = \pm \sqrt{9}$

Step 6

Simplify $\pm \sqrt{9}$ .

Tap for more steps...

Step 6.1

Rewrite $9$ as $3^{2}$ .

$a = \pm \sqrt{3^{2}}$

Step 6.2

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

$a = \pm 3$

Step 7

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

Tap for more steps...

Step 7.1

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

$a = 3$

Step 7.2

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

$a = - 3$

Step 7.3

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

$a = 3, - 3$