Algebra Examples

$\frac{1}{6 a^{2}} - \frac{5 b}{3 a^{3}} + \frac{b^{2}}{4 a}$

Step 1

To write $\frac{1}{6 a^{2}}$ as a fraction with a common denominator, multiply by $\frac{a}{a}$ .

$\frac{1}{6 a^{2}} \cdot \frac{a}{a} - \frac{5 b}{3 a^{3}} + \frac{b^{2}}{4 a}$

Step 2

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

$\frac{1}{6 a^{2}} \cdot \frac{a}{a} - \frac{5 b}{3 a^{3}} \cdot \frac{2}{2} + \frac{b^{2}}{4 a}$

Step 3

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

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Step 3.1

Multiply $\frac{1}{6 a^{2}}$ by $\frac{a}{a}$ .

$\frac{a}{6 a^{2} a} - \frac{5 b}{3 a^{3}} \cdot \frac{2}{2} + \frac{b^{2}}{4 a}$

Step 3.2

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

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

Step 3.3

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

$\frac{a}{6 a^{1 + 2}} - \frac{5 b}{3 a^{3}} \cdot \frac{2}{2} + \frac{b^{2}}{4 a}$

Step 3.4

Add $1$ and $2$ .

$\frac{a}{6 a^{3}} - \frac{5 b}{3 a^{3}} \cdot \frac{2}{2} + \frac{b^{2}}{4 a}$

Step 3.5

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

$\frac{a}{6 a^{3}} - \frac{5 b \cdot 2}{3 a^{3} \cdot 2} + \frac{b^{2}}{4 a}$

Step 3.6

Multiply $2$ by $3$ .

$\frac{a}{6 a^{3}} - \frac{5 b \cdot 2}{6 a^{3}} + \frac{b^{2}}{4 a}$

Step 4

Combine the numerators over the common denominator.

$\frac{a - 5 b \cdot 2}{6 a^{3}} + \frac{b^{2}}{4 a}$

Step 5

Multiply $2$ by $- 5$ .

$\frac{a - 10 b}{6 a^{3}} + \frac{b^{2}}{4 a}$

Step 6

To write $\frac{a - 10 b}{6 a^{3}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{a - 10 b}{6 a^{3}} \cdot \frac{2}{2} + \frac{b^{2}}{4 a}$

Step 7

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

$\frac{a - 10 b}{6 a^{3}} \cdot \frac{2}{2} + \frac{b^{2}}{4 a} \cdot \frac{3 a^{2}}{3 a^{2}}$

Step 8

Write each expression with a common denominator of $12 a^{3}$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.1

Multiply $\frac{a - 10 b}{6 a^{3}}$ by $\frac{2}{2}$ .

$\frac{(a - 10 b) \cdot 2}{6 a^{3} \cdot 2} + \frac{b^{2}}{4 a} \cdot \frac{3 a^{2}}{3 a^{2}}$

Step 8.2

Multiply $2$ by $6$ .

$\frac{(a - 10 b) \cdot 2}{12 a^{3}} + \frac{b^{2}}{4 a} \cdot \frac{3 a^{2}}{3 a^{2}}$

Step 8.3

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

$\frac{(a - 10 b) \cdot 2}{12 a^{3}} + \frac{b^{2} (3 a^{2})}{4 a (3 a^{2})}$

Step 8.4

Multiply $3$ by $4$ .

$\frac{(a - 10 b) \cdot 2}{12 a^{3}} + \frac{b^{2} (3 a^{2})}{12 a \cdot a^{2}}$

Step 8.5

Multiply $a$ by $a^{2}$ by adding the exponents.

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Step 8.5.1

Move $a^{2}$ .

$\frac{(a - 10 b) \cdot 2}{12 a^{3}} + \frac{b^{2} (3 a^{2})}{12 (a^{2} a)}$

Step 8.5.2

Multiply $a^{2}$ by $a$ .

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Step 8.5.2.1

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

$\frac{(a - 10 b) \cdot 2}{12 a^{3}} + \frac{b^{2} (3 a^{2})}{12 (a^{2} a^{1})}$

Step 8.5.2.2

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

$\frac{(a - 10 b) \cdot 2}{12 a^{3}} + \frac{b^{2} (3 a^{2})}{12 a^{2 + 1}}$

Step 8.5.3

Add $2$ and $1$ .

$\frac{(a - 10 b) \cdot 2}{12 a^{3}} + \frac{b^{2} (3 a^{2})}{12 a^{3}}$

Step 9

Combine the numerators over the common denominator.

$\frac{(a - 10 b) \cdot 2 + b^{2} (3 a^{2})}{12 a^{3}}$

Step 10

Simplify the numerator.

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Step 10.1

Apply the distributive property.

$\frac{a \cdot 2 - 10 b \cdot 2 + b^{2} \cdot 3 a^{2}}{12 a^{3}}$

Step 10.2

Move $2$ to the left of $a$ .

$\frac{2 \cdot a - 10 b \cdot 2 + b^{2} \cdot 3 a^{2}}{12 a^{3}}$

Step 10.3

Multiply $2$ by $- 10$ .

$\frac{2 \cdot a - 20 b + b^{2} \cdot 3 a^{2}}{12 a^{3}}$

Step 10.4

Move $3$ to the left of $b^{2}$ .

$\frac{2 a - 20 b + 3 b^{2} a^{2}}{12 a^{3}}$