Precalculus Examples

$\frac{4 a^{3} u}{3 b c^{3}} - \frac{5 v^{2}}{6 a b^{2}}$

Step 1

To write $\frac{4 a^{3} u}{3 b c^{3}}$ as a fraction with a common denominator, multiply by $\frac{2 a b}{2 a b}$ .

$\frac{4 a^{3} u}{3 b c^{3}} \cdot \frac{2 a b}{2 a b} - \frac{5 v^{2}}{6 a b^{2}}$

Step 2

To write $- \frac{5 v^{2}}{6 a b^{2}}$ as a fraction with a common denominator, multiply by $\frac{c^{3}}{c^{3}}$ .

$\frac{4 a^{3} u}{3 b c^{3}} \cdot \frac{2 a b}{2 a b} - \frac{5 v^{2}}{6 a b^{2}} \cdot \frac{c^{3}}{c^{3}}$

Step 3

Write each expression with a common denominator of $6 a b^{2} c^{3}$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{4 a^{3} u}{3 b c^{3}}$ by $\frac{2 a b}{2 a b}$ .

$\frac{4 a^{3} u (2 a b)}{3 b c^{3} (2 a b)} - \frac{5 v^{2}}{6 a b^{2}} \cdot \frac{c^{3}}{c^{3}}$

Step 3.2

Multiply $2$ by $3$ .

$\frac{4 a^{3} u (2 a b)}{6 b c^{3} (a b)} - \frac{5 v^{2}}{6 a b^{2}} \cdot \frac{c^{3}}{c^{3}}$

Step 3.3

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

$\frac{4 a^{3} u (2 a b)}{6 (b^{1} b) c^{3} a} - \frac{5 v^{2}}{6 a b^{2}} \cdot \frac{c^{3}}{c^{3}}$

Step 3.4

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

$\frac{4 a^{3} u (2 a b)}{6 (b^{1} b^{1}) c^{3} a} - \frac{5 v^{2}}{6 a b^{2}} \cdot \frac{c^{3}}{c^{3}}$

Step 3.5

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

$\frac{4 a^{3} u (2 a b)}{6 b^{1 + 1} c^{3} a} - \frac{5 v^{2}}{6 a b^{2}} \cdot \frac{c^{3}}{c^{3}}$

Step 3.6

Add $1$ and $1$ .

$\frac{4 a^{3} u (2 a b)}{6 b^{2} c^{3} a} - \frac{5 v^{2}}{6 a b^{2}} \cdot \frac{c^{3}}{c^{3}}$

Step 3.7

Multiply $\frac{5 v^{2}}{6 a b^{2}}$ by $\frac{c^{3}}{c^{3}}$ .

$\frac{4 a^{3} u (2 a b)}{6 b^{2} c^{3} a} - \frac{5 v^{2} c^{3}}{6 a b^{2} c^{3}}$

Step 3.8

Reorder the factors of $6 b^{2} c^{3} a$ .

$\frac{4 a^{3} u (2 a b)}{6 c^{3} b^{2} a} - \frac{5 v^{2} c^{3}}{6 a b^{2} c^{3}}$

Step 3.9

Reorder the factors of $6 a b^{2} c^{3}$ .

$\frac{4 a^{3} u (2 a b)}{6 c^{3} b^{2} a} - \frac{5 v^{2} c^{3}}{6 c^{3} b^{2} a}$

Step 4

Combine the numerators over the common denominator.

$\frac{4 a^{3} u (2 a b) - 5 v^{2} c^{3}}{6 c^{3} b^{2} a}$

Step 5

Simplify the numerator.

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Step 5.1

Multiply $a^{3}$ by $a$ by adding the exponents.

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Step 5.1.1

Move $a$ .

$\frac{4 (a \cdot a^{3}) u (2 b) - 5 v^{2} c^{3}}{6 c^{3} b^{2} a}$

Step 5.1.2

Multiply $a$ by $a^{3}$ .

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Step 5.1.2.1

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

$\frac{4 (a^{1} a^{3}) u (2 b) - 5 v^{2} c^{3}}{6 c^{3} b^{2} a}$

Step 5.1.2.2

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

$\frac{4 a^{1 + 3} u (2 b) - 5 v^{2} c^{3}}{6 c^{3} b^{2} a}$

Step 5.1.3

Add $1$ and $3$ .

$\frac{4 a^{4} u (2 b) - 5 v^{2} c^{3}}{6 c^{3} b^{2} a}$

Step 5.2

Multiply $2$ by $4$ .

$\frac{8 a^{4} u b - 5 v^{2} c^{3}}{6 c^{3} b^{2} a}$