Pre-Algebra Examples

$\frac{\sqrt{4 a^{2} - 4 b^{2}}}{\sqrt{8 a - 8 b}}$

Step 1

Combine $\sqrt{4 a^{2} - 4 b^{2}}$ and $\sqrt{8 a - 8 b}$ into a single radical.

$\sqrt{\frac{4 a^{2} - 4 b^{2}}{8 a - 8 b}}$

Step 2

Factor $4$ out of $4 a^{2} - 4 b^{2}$ .

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Step 2.1

Factor $4$ out of $4 a^{2}$ .

$\sqrt{\frac{4 (a^{2}) - 4 b^{2}}{8 a - 8 b}}$

Step 2.2

Factor $4$ out of $- 4 b^{2}$ .

$\sqrt{\frac{4 (a^{2}) + 4 (- b^{2})}{8 a - 8 b}}$

Step 2.3

Factor $4$ out of $4 (a^{2}) + 4 (- b^{2})$ .

$\sqrt{\frac{4 (a^{2} - b^{2})}{8 a - 8 b}}$

Step 3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = a$ and $b = b$ .

$\sqrt{\frac{4 (a + b) (a - b)}{8 a - 8 b}}$

Step 4

Factor $8$ out of $8 a - 8 b$ .

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Step 4.1

Factor $8$ out of $8 a$ .

$\sqrt{\frac{4 (a + b) (a - b)}{8 (a) - 8 b}}$

Step 4.2

Factor $8$ out of $- 8 b$ .

$\sqrt{\frac{4 (a + b) (a - b)}{8 (a) + 8 (- b)}}$

Step 4.3

Factor $8$ out of $8 (a) + 8 (- b)$ .

$\sqrt{\frac{4 (a + b) (a - b)}{8 (a - b)}}$

Step 5

Reduce the expression $\frac{4 (a + b) (a - b)}{8 (a - b)}$ by cancelling the common factors.

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Step 5.1

Factor $4$ out of $4 (a + b) (a - b)$ .

$\sqrt{\frac{4 ((a + b) (a - b))}{8 (a - b)}}$

Step 5.2

Factor $4$ out of $8 (a - b)$ .

$\sqrt{\frac{4 ((a + b) (a - b))}{4 (2 (a - b))}}$

Step 5.3

Cancel the common factor.

$\sqrt{\frac{4 ((a + b) (a - b))}{4 (2 (a - b))}}$

Step 5.4

Rewrite the expression.

$\sqrt{\frac{(a + b) (a - b)}{2 (a - b)}}$

Step 6

Cancel the common factor of $a - b$ .

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Step 6.1

Cancel the common factor.

$\sqrt{\frac{(a + b) (a - b)}{2 (a - b)}}$

Step 6.2

Rewrite the expression.

$\sqrt{\frac{a + b}{2}}$

Step 7

Rewrite $\sqrt{\frac{a + b}{2}}$ as $\frac{\sqrt{a + b}}{\sqrt{2}}$ .

$\frac{\sqrt{a + b}}{\sqrt{2}}$

Step 8

Multiply $\frac{\sqrt{a + b}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\sqrt{a + b}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 9

Combine and simplify the denominator.

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Step 9.1

Multiply $\frac{\sqrt{a + b}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\sqrt{a + b} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 9.2

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

$\frac{\sqrt{a + b} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 9.3

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

$\frac{\sqrt{a + b} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 9.4

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

$\frac{\sqrt{a + b} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 9.5

Add $1$ and $1$ .

$\frac{\sqrt{a + b} \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 9.6

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

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Step 9.6.1

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

$\frac{\sqrt{a + b} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 9.6.2

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

$\frac{\sqrt{a + b} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 9.6.3

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

$\frac{\sqrt{a + b} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 9.6.4

Cancel the common factor of $2$ .

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Step 9.6.4.1

Cancel the common factor.

$\frac{\sqrt{a + b} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 9.6.4.2

Rewrite the expression.

$\frac{\sqrt{a + b} \sqrt{2}}{2^{1}}$

Step 9.6.5

Evaluate the exponent.

$\frac{\sqrt{a + b} \sqrt{2}}{2}$

Step 10

Combine using the product rule for radicals.

$\frac{\sqrt{(a + b) \cdot 2}}{2}$

Step 11

Reorder factors in $\frac{\sqrt{(a + b) \cdot 2}}{2}$ .

$\frac{\sqrt{2 (a + b)}}{2}$