Basic Math Examples

$\frac{7 a}{a^{2}} - \frac{a}{4 a + 1}$

Step 1

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

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Step 1.1

Factor $a$ out of $7 a$ .

$\frac{a \cdot 7}{a^{2}} - \frac{a}{4 a + 1}$

Step 1.2

Cancel the common factors.

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Step 1.2.1

Factor $a$ out of $a^{2}$ .

$\frac{a \cdot 7}{a \cdot a} - \frac{a}{4 a + 1}$

Step 1.2.2

Cancel the common factor.

$\frac{a \cdot 7}{a \cdot a} - \frac{a}{4 a + 1}$

Step 1.2.3

Rewrite the expression.

$\frac{7}{a} - \frac{a}{4 a + 1}$

Step 2

To write $\frac{7}{a}$ as a fraction with a common denominator, multiply by $\frac{4 a + 1}{4 a + 1}$ .

$\frac{7}{a} \cdot \frac{4 a + 1}{4 a + 1} - \frac{a}{4 a + 1}$

Step 3

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

$\frac{7}{a} \cdot \frac{4 a + 1}{4 a + 1} - \frac{a}{4 a + 1} \cdot \frac{a}{a}$

Step 4

Write each expression with a common denominator of $a (4 a + 1)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{7}{a}$ by $\frac{4 a + 1}{4 a + 1}$ .

$\frac{7 (4 a + 1)}{a (4 a + 1)} - \frac{a}{4 a + 1} \cdot \frac{a}{a}$

Step 4.2

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

$\frac{7 (4 a + 1)}{a (4 a + 1)} - \frac{a \cdot a}{(4 a + 1) a}$

Step 4.3

Reorder the factors of $(4 a + 1) a$ .

$\frac{7 (4 a + 1)}{a (4 a + 1)} - \frac{a \cdot a}{a (4 a + 1)}$

Step 5

Combine the numerators over the common denominator.

$\frac{7 (4 a + 1) - a \cdot a}{a (4 a + 1)}$

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$\frac{7 (4 a) + 7 \cdot 1 - a \cdot a}{a (4 a + 1)}$

Step 6.2

Multiply $4$ by $7$ .

$\frac{28 a + 7 \cdot 1 - a \cdot a}{a (4 a + 1)}$

Step 6.3

Multiply $7$ by $1$ .

$\frac{28 a + 7 - a \cdot a}{a (4 a + 1)}$

Step 6.4

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

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Step 6.4.1

Move $a$ .

$\frac{28 a + 7 - (a \cdot a)}{a (4 a + 1)}$

Step 6.4.2

Multiply $a$ by $a$ .

$\frac{28 a + 7 - a^{2}}{a (4 a + 1)}$

Step 6.5

Reorder terms.

$\frac{- a^{2} + 28 a + 7}{a (4 a + 1)}$

Step 7

Simplify with factoring out.

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Step 7.1

Factor $- 1$ out of $- a^{2}$ .

$\frac{- (a^{2}) + 28 a + 7}{a (4 a + 1)}$

Step 7.2

Factor $- 1$ out of $28 a$ .

$\frac{- (a^{2}) - (- 28 a) + 7}{a (4 a + 1)}$

Step 7.3

Factor $- 1$ out of $- (a^{2}) - (- 28 a)$ .

$\frac{- (a^{2} - 28 a) + 7}{a (4 a + 1)}$

Step 7.4

Rewrite $7$ as $- 1 (- 7)$ .

$\frac{- (a^{2} - 28 a) - 1 (- 7)}{a (4 a + 1)}$

Step 7.5

Factor $- 1$ out of $- (a^{2} - 28 a) - 1 (- 7)$ .

$\frac{- (a^{2} - 28 a - 7)}{a (4 a + 1)}$

Step 7.6

Simplify the expression.

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Step 7.6.1

Rewrite $- (a^{2} - 28 a - 7)$ as $- 1 (a^{2} - 28 a - 7)$ .

$\frac{- 1 (a^{2} - 28 a - 7)}{a (4 a + 1)}$

Step 7.6.2

Move the negative in front of the fraction.

$- \frac{a^{2} - 28 a - 7}{a (4 a + 1)}$