Algebra Examples

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

Step 1

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

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

Step 2

Combine fractions.

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Step 2.1

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

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

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(\sqrt{a} + 1) (2 \sqrt{a} + \sqrt{b})}{4 {\sqrt{a}}^{2} + 2 \sqrt{b a} - 2 \sqrt{a b} - {\sqrt{b}}^{2}}$

Step 2.3

Simplify.

$\frac{(\sqrt{a} + 1) (2 \sqrt{a} + \sqrt{b})}{4 a - b}$

Step 3

Expand $(\sqrt{a} + 1) (2 \sqrt{a} + \sqrt{b})$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$\frac{\sqrt{a} (2 \sqrt{a} + \sqrt{b}) + 1 (2 \sqrt{a} + \sqrt{b})}{4 a - b}$

Step 3.2

Apply the distributive property.

$\frac{\sqrt{a} (2 \sqrt{a}) + \sqrt{a} \sqrt{b} + 1 (2 \sqrt{a} + \sqrt{b})}{4 a - b}$

Step 3.3

Apply the distributive property.

$\frac{\sqrt{a} (2 \sqrt{a}) + \sqrt{a} \sqrt{b} + 1 (2 \sqrt{a}) + 1 \sqrt{b}}{4 a - b}$

Step 4

Simplify each term.

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Step 4.1

Rewrite using the commutative property of multiplication.

$\frac{2 \sqrt{a} \sqrt{a} + \sqrt{a} \sqrt{b} + 1 (2 \sqrt{a}) + 1 \sqrt{b}}{4 a - b}$

Step 4.2

Multiply $2 \sqrt{a} \sqrt{a}$ .

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Step 4.2.1

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

$\frac{2 ({\sqrt{a}}^{1} \sqrt{a}) + \sqrt{a} \sqrt{b} + 1 (2 \sqrt{a}) + 1 \sqrt{b}}{4 a - b}$

Step 4.2.2

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

$\frac{2 ({\sqrt{a}}^{1} {\sqrt{a}}^{1}) + \sqrt{a} \sqrt{b} + 1 (2 \sqrt{a}) + 1 \sqrt{b}}{4 a - b}$

Step 4.2.3

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

$\frac{2 {\sqrt{a}}^{1 + 1} + \sqrt{a} \sqrt{b} + 1 (2 \sqrt{a}) + 1 \sqrt{b}}{4 a - b}$

Step 4.2.4

Add $1$ and $1$ .

$\frac{2 {\sqrt{a}}^{2} + \sqrt{a} \sqrt{b} + 1 (2 \sqrt{a}) + 1 \sqrt{b}}{4 a - b}$

Step 4.3

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

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Step 4.3.1

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

$\frac{2 {(a^{\frac{1}{2}})}^{2} + \sqrt{a} \sqrt{b} + 1 (2 \sqrt{a}) + 1 \sqrt{b}}{4 a - b}$

Step 4.3.2

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

$\frac{2 a^{\frac{1}{2} \cdot 2} + \sqrt{a} \sqrt{b} + 1 (2 \sqrt{a}) + 1 \sqrt{b}}{4 a - b}$

Step 4.3.3

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

$\frac{2 a^{\frac{2}{2}} + \sqrt{a} \sqrt{b} + 1 (2 \sqrt{a}) + 1 \sqrt{b}}{4 a - b}$

Step 4.3.4

Cancel the common factor of $2$ .

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Step 4.3.4.1

Cancel the common factor.

$\frac{2 a^{\frac{2}{2}} + \sqrt{a} \sqrt{b} + 1 (2 \sqrt{a}) + 1 \sqrt{b}}{4 a - b}$

Step 4.3.4.2

Rewrite the expression.

$\frac{2 a^{1} + \sqrt{a} \sqrt{b} + 1 (2 \sqrt{a}) + 1 \sqrt{b}}{4 a - b}$

Step 4.3.5

Simplify.

$\frac{2 a + \sqrt{a} \sqrt{b} + 1 (2 \sqrt{a}) + 1 \sqrt{b}}{4 a - b}$

Step 4.4

Combine using the product rule for radicals.

$\frac{2 a + \sqrt{a b} + 1 (2 \sqrt{a}) + 1 \sqrt{b}}{4 a - b}$

Step 4.5

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

$\frac{2 a + \sqrt{a b} + 2 \sqrt{a} + 1 \sqrt{b}}{4 a - b}$

Step 4.6

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

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

Step 5

Simplify the numerator.

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Step 5.1

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

$\frac{2 a + {(a b)}^{\frac{1}{2}} + 2 \sqrt{a} + \sqrt{b}}{4 a - b}$

Step 5.2

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

$\frac{2 a + {(a b)}^{\frac{1}{2}} + 2 a^{\frac{1}{2}} + \sqrt{b}}{4 a - b}$

Step 5.3

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

$\frac{2 a + {(a b)}^{\frac{1}{2}} + 2 a^{\frac{1}{2}} + b^{\frac{1}{2}}}{4 a - b}$

Step 5.4

Apply the product rule to $a b$ .

$\frac{2 a + a^{\frac{1}{2}} b^{\frac{1}{2}} + 2 a^{\frac{1}{2}} + b^{\frac{1}{2}}}{4 a - b}$

Step 5.5

Rewrite $2 a + a^{\frac{1}{2}} b^{\frac{1}{2}} + 2 a^{\frac{1}{2}} + b^{\frac{1}{2}}$ in a factored form.

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Step 5.5.1

Factor $a^{\frac{1}{2}}$ out of $2 a + a^{\frac{1}{2}} b^{\frac{1}{2}}$ .

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Step 5.5.1.1

Factor $a^{\frac{1}{2}}$ out of $2 a$ .

$\frac{a^{\frac{1}{2}} (2 a^{\frac{1}{2}}) + a^{\frac{1}{2}} b^{\frac{1}{2}} + 2 a^{\frac{1}{2}} + b^{\frac{1}{2}}}{4 a - b}$

Step 5.5.1.2

Factor $a^{\frac{1}{2}}$ out of $a^{\frac{1}{2}} b^{\frac{1}{2}}$ .

$\frac{a^{\frac{1}{2}} (2 a^{\frac{1}{2}}) + a^{\frac{1}{2}} (b^{\frac{1}{2}}) + 2 a^{\frac{1}{2}} + b^{\frac{1}{2}}}{4 a - b}$

Step 5.5.1.3

Factor $a^{\frac{1}{2}}$ out of $a^{\frac{1}{2}} (2 a^{\frac{1}{2}}) + a^{\frac{1}{2}} (b^{\frac{1}{2}})$ .

$\frac{a^{\frac{1}{2}} (2 a^{\frac{1}{2}} + b^{\frac{1}{2}}) + 2 a^{\frac{1}{2}} + b^{\frac{1}{2}}}{4 a - b}$

Step 5.5.2

Let $u = a^{\frac{1}{2}}$ . Substitute $u$ for all occurrences of $a^{\frac{1}{2}}$ .

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Step 5.5.2.1

Apply the distributive property.

$\frac{u (2 u) + u b^{\frac{1}{2}} + 2 u + b^{\frac{1}{2}}}{4 a - b}$

Step 5.5.2.2

Rewrite using the commutative property of multiplication.

$\frac{2 u \cdot u + u b^{\frac{1}{2}} + 2 u + b^{\frac{1}{2}}}{4 a - b}$

Step 5.5.2.3

Multiply $u$ by $u$ by adding the exponents.

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Step 5.5.2.3.1

Move $u$ .

$\frac{2 (u \cdot u) + u b^{\frac{1}{2}} + 2 u + b^{\frac{1}{2}}}{4 a - b}$

Step 5.5.2.3.2

Multiply $u$ by $u$ .

$\frac{2 u^{2} + u b^{\frac{1}{2}} + 2 u + b^{\frac{1}{2}}}{4 a - b}$

Step 5.5.3

Factor out the greatest common factor from each group.

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Step 5.5.3.1

Group the first two terms and the last two terms.

$\frac{(2 u^{2} + u b^{\frac{1}{2}}) + 2 u + b^{\frac{1}{2}}}{4 a - b}$

Step 5.5.3.2

Factor out the greatest common factor (GCF) from each group.

$\frac{u (2 u + b^{\frac{1}{2}}) + 1 (2 u + b^{\frac{1}{2}})}{4 a - b}$

Step 5.5.4

Factor the polynomial by factoring out the greatest common factor, $2 u + b^{\frac{1}{2}}$ .

$\frac{(2 u + b^{\frac{1}{2}}) (u + 1)}{4 a - b}$

Step 5.5.5

Replace all occurrences of $u$ with $a^{\frac{1}{2}}$ .

$\frac{(2 a^{\frac{1}{2}} + b^{\frac{1}{2}}) (a^{\frac{1}{2}} + 1)}{4 a - b}$