Algebra Examples

$\frac{5 b}{12 a^{2}} + \frac{1}{3 a} - \frac{3 b^{2}}{8 a^{3}}$

Step 1

To write $\frac{1}{3 a}$ as a fraction with a common denominator, multiply by $\frac{4 a}{4 a}$ .

$\frac{5 b}{12 a^{2}} + \frac{1}{3 a} \cdot \frac{4 a}{4 a} - \frac{3 b^{2}}{8 a^{3}}$

Step 2

Write each expression with a common denominator of $12 a^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.1

Multiply $\frac{1}{3 a}$ by $\frac{4 a}{4 a}$ .

$\frac{5 b}{12 a^{2}} + \frac{4 a}{3 a (4 a)} - \frac{3 b^{2}}{8 a^{3}}$

Step 2.2

Multiply $4$ by $3$ .

$\frac{5 b}{12 a^{2}} + \frac{4 a}{12 a \cdot a} - \frac{3 b^{2}}{8 a^{3}}$

Step 2.3

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

$\frac{5 b}{12 a^{2}} + \frac{4 a}{12 (a^{1} a)} - \frac{3 b^{2}}{8 a^{3}}$

Step 2.4

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

$\frac{5 b}{12 a^{2}} + \frac{4 a}{12 (a^{1} a^{1})} - \frac{3 b^{2}}{8 a^{3}}$

Step 2.5

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

$\frac{5 b}{12 a^{2}} + \frac{4 a}{12 a^{1 + 1}} - \frac{3 b^{2}}{8 a^{3}}$

Step 2.6

Add $1$ and $1$ .

$\frac{5 b}{12 a^{2}} + \frac{4 a}{12 a^{2}} - \frac{3 b^{2}}{8 a^{3}}$

Step 3

Combine the numerators over the common denominator.

$\frac{5 b + 4 a}{12 a^{2}} - \frac{3 b^{2}}{8 a^{3}}$

Step 4

To write $\frac{5 b + 4 a}{12 a^{2}}$ as a fraction with a common denominator, multiply by $\frac{2 a}{2 a}$ .

$\frac{5 b + 4 a}{12 a^{2}} \cdot \frac{2 a}{2 a} - \frac{3 b^{2}}{8 a^{3}}$

Step 5

To write $- \frac{3 b^{2}}{8 a^{3}}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{5 b + 4 a}{12 a^{2}} \cdot \frac{2 a}{2 a} - \frac{3 b^{2}}{8 a^{3}} \cdot \frac{3}{3}$

Step 6

Write each expression with a common denominator of $24 a^{3}$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.1

Multiply $\frac{5 b + 4 a}{12 a^{2}}$ by $\frac{2 a}{2 a}$ .

$\frac{(5 b + 4 a) (2 a)}{12 a^{2} (2 a)} - \frac{3 b^{2}}{8 a^{3}} \cdot \frac{3}{3}$

Step 6.2

Multiply $2$ by $12$ .

$\frac{(5 b + 4 a) (2 a)}{24 a^{2} a} - \frac{3 b^{2}}{8 a^{3}} \cdot \frac{3}{3}$

Step 6.3

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

$\frac{(5 b + 4 a) (2 a)}{24 (a^{1} a^{2})} - \frac{3 b^{2}}{8 a^{3}} \cdot \frac{3}{3}$

Step 6.4

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

$\frac{(5 b + 4 a) (2 a)}{24 a^{1 + 2}} - \frac{3 b^{2}}{8 a^{3}} \cdot \frac{3}{3}$

Step 6.5

Add $1$ and $2$ .

$\frac{(5 b + 4 a) (2 a)}{24 a^{3}} - \frac{3 b^{2}}{8 a^{3}} \cdot \frac{3}{3}$

Step 6.6

Multiply $\frac{3 b^{2}}{8 a^{3}}$ by $\frac{3}{3}$ .

$\frac{(5 b + 4 a) (2 a)}{24 a^{3}} - \frac{3 b^{2} \cdot 3}{8 a^{3} \cdot 3}$

Step 6.7

Multiply $3$ by $8$ .

$\frac{(5 b + 4 a) (2 a)}{24 a^{3}} - \frac{3 b^{2} \cdot 3}{24 a^{3}}$

Step 7

Combine the numerators over the common denominator.

$\frac{(5 b + 4 a) (2 a) - 3 b^{2} \cdot 3}{24 a^{3}}$

Step 8

Simplify the numerator.

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Step 8.1

Apply the distributive property.

$\frac{(5 b \cdot 2 + 4 a \cdot 2) a - 3 b^{2} \cdot 3}{24 a^{3}}$

Step 8.2

Multiply $2$ by $5$ .

$\frac{(10 b + 4 a \cdot 2) a - 3 b^{2} \cdot 3}{24 a^{3}}$

Step 8.3

Multiply $2$ by $4$ .

$\frac{(10 b + 8 a) a - 3 b^{2} \cdot 3}{24 a^{3}}$

Step 8.4

Apply the distributive property.

$\frac{10 b a + 8 a \cdot a - 3 b^{2} \cdot 3}{24 a^{3}}$

Step 8.5

Multiply $a$ by $a$ by adding the exponents.

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Step 8.5.1

Move $a$ .

$\frac{10 b a + 8 (a \cdot a) - 3 b^{2} \cdot 3}{24 a^{3}}$

Step 8.5.2

Multiply $a$ by $a$ .

$\frac{10 b a + 8 a^{2} - 3 b^{2} \cdot 3}{24 a^{3}}$

Step 8.6

Multiply $3$ by $- 3$ .

$\frac{10 b a + 8 a^{2} - 9 b^{2}}{24 a^{3}}$