Precalculus Examples

$\frac{2}{a^{2}} - \frac{3}{a b} + \frac{4}{b^{2}}$

Step 1

To write $\frac{2}{a^{2}}$ as a fraction with a common denominator, multiply by $\frac{b}{b}$ .

$\frac{2}{a^{2}} \cdot \frac{b}{b} - \frac{3}{a b} + \frac{4}{b^{2}}$

Step 2

To write $- \frac{3}{a b}$ as a fraction with a common denominator, multiply by $\frac{a}{a}$ .

$\frac{2}{a^{2}} \cdot \frac{b}{b} - \frac{3}{a b} \cdot \frac{a}{a} + \frac{4}{b^{2}}$

Step 3

Write each expression with a common denominator of $b a^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{2}{a^{2}}$ by $\frac{b}{b}$ .

$\frac{2 b}{a^{2} b} - \frac{3}{a b} \cdot \frac{a}{a} + \frac{4}{b^{2}}$

Step 3.2

Multiply $\frac{3}{a b}$ by $\frac{a}{a}$ .

$\frac{2 b}{a^{2} b} - \frac{3 a}{a b a} + \frac{4}{b^{2}}$

Step 3.3

Raise $a$ to the power of $1$ .

$\frac{2 b}{a^{2} b} - \frac{3 a}{a^{1} a b} + \frac{4}{b^{2}}$

Step 3.4

Raise $a$ to the power of $1$ .

$\frac{2 b}{a^{2} b} - \frac{3 a}{a^{1} a^{1} b} + \frac{4}{b^{2}}$

Step 3.5

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

$\frac{2 b}{a^{2} b} - \frac{3 a}{a^{1 + 1} b} + \frac{4}{b^{2}}$

Step 3.6

Add $1$ and $1$ .

$\frac{2 b}{a^{2} b} - \frac{3 a}{a^{2} b} + \frac{4}{b^{2}}$

Step 4

Combine the numerators over the common denominator.

$\frac{2 b - 3 a}{a^{2} b} + \frac{4}{b^{2}}$

Step 5

To write $\frac{2 b - 3 a}{a^{2} b}$ as a fraction with a common denominator, multiply by $\frac{b}{b}$ .

$\frac{2 b - 3 a}{a^{2} b} \cdot \frac{b}{b} + \frac{4}{b^{2}}$

Step 6

To write $\frac{4}{b^{2}}$ as a fraction with a common denominator, multiply by $\frac{a^{2}}{a^{2}}$ .

$\frac{2 b - 3 a}{a^{2} b} \cdot \frac{b}{b} + \frac{4}{b^{2}} \cdot \frac{a^{2}}{a^{2}}$

Step 7

Write each expression with a common denominator of $a^{2} b^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.1

Multiply $\frac{2 b - 3 a}{a^{2} b}$ by $\frac{b}{b}$ .

$\frac{(2 b - 3 a) b}{a^{2} b \cdot b} + \frac{4}{b^{2}} \cdot \frac{a^{2}}{a^{2}}$

Step 7.2

Raise $b$ to the power of $1$ .

$\frac{(2 b - 3 a) b}{a^{2} (b^{1} b)} + \frac{4}{b^{2}} \cdot \frac{a^{2}}{a^{2}}$

Step 7.3

Raise $b$ to the power of $1$ .

$\frac{(2 b - 3 a) b}{a^{2} (b^{1} b^{1})} + \frac{4}{b^{2}} \cdot \frac{a^{2}}{a^{2}}$

Step 7.4

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

$\frac{(2 b - 3 a) b}{a^{2} b^{1 + 1}} + \frac{4}{b^{2}} \cdot \frac{a^{2}}{a^{2}}$

Step 7.5

Add $1$ and $1$ .

$\frac{(2 b - 3 a) b}{a^{2} b^{2}} + \frac{4}{b^{2}} \cdot \frac{a^{2}}{a^{2}}$

Step 7.6

Multiply $\frac{4}{b^{2}}$ by $\frac{a^{2}}{a^{2}}$ .

$\frac{(2 b - 3 a) b}{a^{2} b^{2}} + \frac{4 a^{2}}{b^{2} a^{2}}$

Step 7.7

Reorder the factors of $b^{2} a^{2}$ .

$\frac{(2 b - 3 a) b}{a^{2} b^{2}} + \frac{4 a^{2}}{a^{2} b^{2}}$

Step 8

Combine the numerators over the common denominator.

$\frac{(2 b - 3 a) b + 4 a^{2}}{a^{2} b^{2}}$

Step 9

Simplify the numerator.

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Step 9.1

Apply the distributive property.

$\frac{2 b \cdot b - 3 a b + 4 a^{2}}{a^{2} b^{2}}$

Step 9.2

Multiply $b$ by $b$ by adding the exponents.

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Step 9.2.1

Move $b$ .

$\frac{2 (b \cdot b) - 3 a b + 4 a^{2}}{a^{2} b^{2}}$

Step 9.2.2

Multiply $b$ by $b$ .

$\frac{2 b^{2} - 3 a b + 4 a^{2}}{a^{2} b^{2}}$