Pre-Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

Expand the denominator using the FOIL method.

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

Step 1.4

Simplify.

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

Step 1.5

Simplify the numerator.

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Step 1.5.1

Raise $\sqrt{a} - \sqrt{b}$ to the power of $1$ .

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

Step 1.5.2

Raise $\sqrt{a} - \sqrt{b}$ to the power of $1$ .

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

Step 1.5.3

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

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

Step 1.5.4

Add $1$ and $1$ .

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

Step 1.6

Rewrite ${(\sqrt{a} - \sqrt{b})}^{2}$ as $(\sqrt{a} - \sqrt{b}) (\sqrt{a} - \sqrt{b})$ .

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

Step 1.7

Expand $(\sqrt{a} - \sqrt{b}) (\sqrt{a} - \sqrt{b})$ using the FOIL Method.

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Step 1.7.1

Apply the distributive property.

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

Step 1.7.2

Apply the distributive property.

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

Step 1.7.3

Apply the distributive property.

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

Step 1.8

Simplify and combine like terms.

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Step 1.8.1

Simplify each term.

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Step 1.8.1.1

Multiply $\sqrt{a} \sqrt{a}$ .

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Step 1.8.1.1.1

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

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

Step 1.8.1.1.2

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

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

Step 1.8.1.1.3

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

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

Step 1.8.1.1.4

Add $1$ and $1$ .

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

Step 1.8.1.2

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

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Step 1.8.1.2.1

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

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

Step 1.8.1.2.2

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

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

Step 1.8.1.2.3

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

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

Step 1.8.1.2.4

Cancel the common factor of $2$ .

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Step 1.8.1.2.4.1

Cancel the common factor.

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

Step 1.8.1.2.4.2

Rewrite the expression.

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

Step 1.8.1.2.5

Simplify.

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

Step 1.8.1.3

Rewrite using the commutative property of multiplication.

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

Step 1.8.1.4

Combine using the product rule for radicals.

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

Step 1.8.1.5

Combine using the product rule for radicals.

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

Step 1.8.1.6

Multiply $- \sqrt{b} (- \sqrt{b})$ .

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Step 1.8.1.6.1

Multiply $- 1$ by $- 1$ .

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

Step 1.8.1.6.2

Multiply $\sqrt{b}$ by $1$ .

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

Step 1.8.1.6.3

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

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

Step 1.8.1.6.4

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

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

Step 1.8.1.6.5

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

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

Step 1.8.1.6.6

Add $1$ and $1$ .

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

Step 1.8.1.7

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

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Step 1.8.1.7.1

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

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

Step 1.8.1.7.2

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

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

Step 1.8.1.7.3

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

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

Step 1.8.1.7.4

Cancel the common factor of $2$ .

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Step 1.8.1.7.4.1

Cancel the common factor.

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

Step 1.8.1.7.4.2

Rewrite the expression.

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

Step 1.8.1.7.5

Simplify.

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

Step 1.8.2

Subtract $\sqrt{a b}$ from $- \sqrt{b a}$ .

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Step 1.8.2.1

Reorder $b$ and $a$ .

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

Step 1.8.2.2

Subtract $\sqrt{a b}$ from $- \sqrt{a b}$ .

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

Step 2

Simplify terms.

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Step 2.1

Combine the numerators over the common denominator.

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

Step 2.2

Combine the opposite terms in $a - 2 \sqrt{a b} + b + a + 2 \sqrt{a b} + b$ .

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Step 2.2.1

Add $- 2 \sqrt{a b}$ and $2 \sqrt{a b}$ .

$\frac{a + b + a + 0 + b}{a - b}$

Step 2.2.2

Add $a + b + a$ and $0$ .

$\frac{a + b + a + b}{a - b}$

Step 2.3

Add $a$ and $a$ .

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

Step 2.4

Add $b$ and $b$ .

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

Step 2.5

Factor $2$ out of $2 a + 2 b$ .

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Step 2.5.1

Factor $2$ out of $2 a$ .

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

Step 2.5.2

Factor $2$ out of $2 b$ .

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

Step 2.5.3

Factor $2$ out of $2 (a) + 2 (b)$ .

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