Basic Math Examples

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

Step 1

Simplify each term.

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Step 1.1

To divide by a fraction, multiply by its reciprocal.

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

Step 1.2

Combine.

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

Step 1.3

Combine.

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

Step 1.4

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

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Step 1.4.1

Move $q$ .

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

Step 1.4.2

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

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Step 1.4.2.1

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

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

Step 1.4.2.2

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

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

Step 1.4.3

Add $1$ and $2$ .

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

Step 1.5

Multiply $q$ by $q$ by adding the exponents.

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Step 1.5.1

Move $q$ .

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

Step 1.5.2

Multiply $q$ by $q$ .

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

Step 1.6

Cancel the common factor of $2$ and $4$ .

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Step 1.6.1

Factor $2$ out of $2 q^{3} (a c) \cdot 3$ .

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

Step 1.6.2

Cancel the common factors.

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Step 1.6.2.1

Factor $2$ out of $3 a^{2} (4 q^{2})$ .

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

Step 1.6.2.2

Cancel the common factor.

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

Step 1.6.2.3

Rewrite the expression.

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

Step 1.7

Cancel the common factor of $q^{3}$ and $q^{2}$ .

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Step 1.7.1

Factor $q^{2}$ out of $(q^{3} (a c)) \cdot 3$ .

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

Step 1.7.2

Cancel the common factors.

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Step 1.7.2.1

Factor $q^{2}$ out of $3 a^{2} (2 q^{2})$ .

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

Step 1.7.2.2

Cancel the common factor.

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

Step 1.7.2.3

Rewrite the expression.

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

Step 1.8

Cancel the common factor of $a$ and $a^{2}$ .

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Step 1.8.1

Factor $a$ out of $(q (a c)) \cdot 3$ .

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

Step 1.8.2

Cancel the common factors.

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Step 1.8.2.1

Factor $a$ out of $3 a^{2} \cdot (2)$ .

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

Step 1.8.2.2

Cancel the common factor.

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

Step 1.8.2.3

Rewrite the expression.

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

Step 1.9

Cancel the common factor of $3$ .

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Step 1.9.1

Cancel the common factor.

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

Step 1.9.2

Rewrite the expression.

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

Step 1.10

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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Step 4.1

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

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

Step 4.2

Multiply $\frac{q c}{2 a}$ by $\frac{b}{b}$ .

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

Step 4.3

Reorder the factors of $2 b a$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

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

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Step 6.1

Move $a$ .

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

Step 6.2

Multiply $a$ by $a$ .

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

Step 7

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

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

Step 8

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

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

Step 9

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

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Step 9.1

Multiply $\frac{3 a^{2} + q c b}{2 a b}$ by $\frac{a}{a}$ .

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

Step 9.2

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

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

Step 9.3

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

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

Step 9.4

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

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

Step 9.5

Add $1$ and $1$ .

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

Step 9.6

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

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

Step 9.7

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

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

Step 10

Combine the numerators over the common denominator.

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

Step 11

Simplify the numerator.

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Step 11.1

Apply the distributive property.

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

Step 11.2

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

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Step 11.2.1

Move $a$ .

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

Step 11.2.2

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

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Step 11.2.2.1

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

$\frac{3 (a^{1} a^{2}) + q c b a - 2 b (2 b)}{2 a^{2} b}$

Step 11.2.2.2

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

$\frac{3 a^{1 + 2} + q c b a - 2 b (2 b)}{2 a^{2} b}$

Step 11.2.3

Add $1$ and $2$ .

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

Step 11.3

Rewrite using the commutative property of multiplication.

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

Step 11.4

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

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Step 11.4.1

Move $b$ .

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

Step 11.4.2

Multiply $b$ by $b$ .

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

Step 11.5

Multiply $- 2$ by $2$ .

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