Algebra Examples

$\frac{d + 3}{8 d} - \frac{2 d + 1}{10 d^{2}}$

Step 1

To write $\frac{d + 3}{8 d}$ as a fraction with a common denominator, multiply by $\frac{5 d}{5 d}$ .

$\frac{d + 3}{8 d} \cdot \frac{5 d}{5 d} - \frac{2 d + 1}{10 d^{2}}$

Step 2

To write $- \frac{2 d + 1}{10 d^{2}}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{d + 3}{8 d} \cdot \frac{5 d}{5 d} - \frac{2 d + 1}{10 d^{2}} \cdot \frac{4}{4}$

Step 3

Write each expression with a common denominator of $40 d^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{d + 3}{8 d}$ by $\frac{5 d}{5 d}$ .

$\frac{(d + 3) (5 d)}{8 d (5 d)} - \frac{2 d + 1}{10 d^{2}} \cdot \frac{4}{4}$

Step 3.2

Multiply $5$ by $8$ .

$\frac{(d + 3) (5 d)}{40 d \cdot d} - \frac{2 d + 1}{10 d^{2}} \cdot \frac{4}{4}$

Step 3.3

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

$\frac{(d + 3) (5 d)}{40 (d^{1} d)} - \frac{2 d + 1}{10 d^{2}} \cdot \frac{4}{4}$

Step 3.4

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

$\frac{(d + 3) (5 d)}{40 (d^{1} d^{1})} - \frac{2 d + 1}{10 d^{2}} \cdot \frac{4}{4}$

Step 3.5

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

$\frac{(d + 3) (5 d)}{40 d^{1 + 1}} - \frac{2 d + 1}{10 d^{2}} \cdot \frac{4}{4}$

Step 3.6

Add $1$ and $1$ .

$\frac{(d + 3) (5 d)}{40 d^{2}} - \frac{2 d + 1}{10 d^{2}} \cdot \frac{4}{4}$

Step 3.7

Multiply $\frac{2 d + 1}{10 d^{2}}$ by $\frac{4}{4}$ .

$\frac{(d + 3) (5 d)}{40 d^{2}} - \frac{(2 d + 1) \cdot 4}{10 d^{2} \cdot 4}$

Step 3.8

Multiply $4$ by $10$ .

$\frac{(d + 3) (5 d)}{40 d^{2}} - \frac{(2 d + 1) \cdot 4}{40 d^{2}}$

Step 4

Combine the numerators over the common denominator.

$\frac{(d + 3) (5 d) - (2 d + 1) \cdot 4}{40 d^{2}}$

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

$\frac{(d \cdot 5 + 3 \cdot 5) d - (2 d + 1) \cdot 4}{40 d^{2}}$

Step 5.2

Move $5$ to the left of $d$ .

$\frac{(5 \cdot d + 3 \cdot 5) d - (2 d + 1) \cdot 4}{40 d^{2}}$

Step 5.3

Multiply $3$ by $5$ .

$\frac{(5 \cdot d + 15) d - (2 d + 1) \cdot 4}{40 d^{2}}$

Step 5.4

Apply the distributive property.

$\frac{5 d \cdot d + 15 d - (2 d + 1) \cdot 4}{40 d^{2}}$

Step 5.5

Multiply $d$ by $d$ by adding the exponents.

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Step 5.5.1

Move $d$ .

$\frac{5 (d \cdot d) + 15 d - (2 d + 1) \cdot 4}{40 d^{2}}$

Step 5.5.2

Multiply $d$ by $d$ .

$\frac{5 d^{2} + 15 d - (2 d + 1) \cdot 4}{40 d^{2}}$

Step 5.6

Apply the distributive property.

$\frac{5 d^{2} + 15 d + (- (2 d) - 1 \cdot 1) \cdot 4}{40 d^{2}}$

Step 5.7

Multiply $2$ by $- 1$ .

$\frac{5 d^{2} + 15 d + (- 2 d - 1 \cdot 1) \cdot 4}{40 d^{2}}$

Step 5.8

Multiply $- 1$ by $1$ .

$\frac{5 d^{2} + 15 d + (- 2 d - 1) \cdot 4}{40 d^{2}}$

Step 5.9

Apply the distributive property.

$\frac{5 d^{2} + 15 d - 2 d \cdot 4 - 1 \cdot 4}{40 d^{2}}$

Step 5.10

Multiply $4$ by $- 2$ .

$\frac{5 d^{2} + 15 d - 8 d - 1 \cdot 4}{40 d^{2}}$

Step 5.11

Multiply $- 1$ by $4$ .

$\frac{5 d^{2} + 15 d - 8 d - 4}{40 d^{2}}$

Step 5.12

Subtract $8 d$ from $15 d$ .

$\frac{5 d^{2} + 7 d - 4}{40 d^{2}}$