Algebra Examples

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

Step 1

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

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

Step 2

To write $\frac{5 b}{3 a}$ as a fraction with a common denominator, multiply by $\frac{4 a}{4 a}$ .

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

Step 3

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 3.1

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

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

Step 3.2

Multiply $3$ by $4$ .

$\frac{3 b^{2} \cdot 3}{12 a^{2}} + \frac{5 b}{3 a} \cdot \frac{4 a}{4 a} - \frac{1}{6 a^{3}}$

Step 3.3

Multiply $\frac{5 b}{3 a}$ by $\frac{4 a}{4 a}$ .

$\frac{3 b^{2} \cdot 3}{12 a^{2}} + \frac{5 b (4 a)}{3 a (4 a)} - \frac{1}{6 a^{3}}$

Step 3.4

Multiply $4$ by $3$ .

$\frac{3 b^{2} \cdot 3}{12 a^{2}} + \frac{5 b (4 a)}{12 a \cdot a} - \frac{1}{6 a^{3}}$

Step 3.5

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

$\frac{3 b^{2} \cdot 3}{12 a^{2}} + \frac{5 b (4 a)}{12 (a^{1} a)} - \frac{1}{6 a^{3}}$

Step 3.6

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

$\frac{3 b^{2} \cdot 3}{12 a^{2}} + \frac{5 b (4 a)}{12 (a^{1} a^{1})} - \frac{1}{6 a^{3}}$

Step 3.7

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

$\frac{3 b^{2} \cdot 3}{12 a^{2}} + \frac{5 b (4 a)}{12 a^{1 + 1}} - \frac{1}{6 a^{3}}$

Step 3.8

Add $1$ and $1$ .

$\frac{3 b^{2} \cdot 3}{12 a^{2}} + \frac{5 b (4 a)}{12 a^{2}} - \frac{1}{6 a^{3}}$

Step 4

Combine the numerators over the common denominator.

$\frac{3 b^{2} \cdot 3 + 5 b (4 a)}{12 a^{2}} - \frac{1}{6 a^{3}}$

Step 5

Simplify the numerator.

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Step 5.1

Factor $b$ out of $3 b^{2} \cdot 3 + 5 b \cdot 4 a$ .

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Step 5.1.1

Factor $b$ out of $3 b^{2} \cdot 3$ .

$\frac{b (3 b \cdot 3) + 5 b \cdot 4 a}{12 a^{2}} - \frac{1}{6 a^{3}}$

Step 5.1.2

Factor $b$ out of $5 b \cdot 4 a$ .

$\frac{b (3 b \cdot 3) + b (5 \cdot 4 a)}{12 a^{2}} - \frac{1}{6 a^{3}}$

Step 5.1.3

Factor $b$ out of $b (3 b \cdot 3) + b (5 \cdot 4 a)$ .

$\frac{b (3 b \cdot 3 + 5 \cdot 4 a)}{12 a^{2}} - \frac{1}{6 a^{3}}$

Step 5.2

Multiply $3$ by $3$ .

$\frac{b (9 b + 5 \cdot 4 a)}{12 a^{2}} - \frac{1}{6 a^{3}}$

Step 5.3

Multiply $5$ by $4$ .

$\frac{b (9 b + 20 a)}{12 a^{2}} - \frac{1}{6 a^{3}}$

Step 6

To write $\frac{b (9 b + 20 a)}{12 a^{2}}$ as a fraction with a common denominator, multiply by $\frac{a}{a}$ .

$\frac{b (9 b + 20 a)}{12 a^{2}} \cdot \frac{a}{a} - \frac{1}{6 a^{3}}$

Step 7

To write $- \frac{1}{6 a^{3}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{b (9 b + 20 a)}{12 a^{2}} \cdot \frac{a}{a} - \frac{1}{6 a^{3}} \cdot \frac{2}{2}$

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 $\frac{b (9 b + 20 a)}{12 a^{2}}$ by $\frac{a}{a}$ .

$\frac{b (9 b + 20 a) a}{12 a^{2} a} - \frac{1}{6 a^{3}} \cdot \frac{2}{2}$

Step 8.2

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

$\frac{b (9 b + 20 a) a}{12 (a^{1} a^{2})} - \frac{1}{6 a^{3}} \cdot \frac{2}{2}$

Step 8.3

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

$\frac{b (9 b + 20 a) a}{12 a^{1 + 2}} - \frac{1}{6 a^{3}} \cdot \frac{2}{2}$

Step 8.4

Add $1$ and $2$ .

$\frac{b (9 b + 20 a) a}{12 a^{3}} - \frac{1}{6 a^{3}} \cdot \frac{2}{2}$

Step 8.5

Multiply $\frac{1}{6 a^{3}}$ by $\frac{2}{2}$ .

$\frac{b (9 b + 20 a) a}{12 a^{3}} - \frac{2}{6 a^{3} \cdot 2}$

Step 8.6

Multiply $2$ by $6$ .

$\frac{b (9 b + 20 a) a}{12 a^{3}} - \frac{2}{12 a^{3}}$

Step 9

Combine the numerators over the common denominator.

$\frac{b (9 b + 20 a) a - 2}{12 a^{3}}$

Step 10

Simplify the numerator.

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Step 10.1

Apply the distributive property.

$\frac{(b (9 b) + b (20 a)) a - 2}{12 a^{3}}$

Step 10.2

Rewrite using the commutative property of multiplication.

$\frac{(9 b \cdot b + b (20 a)) a - 2}{12 a^{3}}$

Step 10.3

Rewrite using the commutative property of multiplication.

$\frac{(9 b \cdot b + 20 b a) a - 2}{12 a^{3}}$

Step 10.4

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

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Step 10.4.1

Move $b$ .

$\frac{(9 (b \cdot b) + 20 b a) a - 2}{12 a^{3}}$

Step 10.4.2

Multiply $b$ by $b$ .

$\frac{(9 b^{2} + 20 b a) a - 2}{12 a^{3}}$

Step 10.5

Apply the distributive property.

$\frac{9 b^{2} a + 20 b a \cdot a - 2}{12 a^{3}}$

Step 10.6

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

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Step 10.6.1

Move $a$ .

$\frac{9 b^{2} a + 20 b (a \cdot a) - 2}{12 a^{3}}$

Step 10.6.2

Multiply $a$ by $a$ .

$\frac{9 b^{2} a + 20 b a^{2} - 2}{12 a^{3}}$