Basic Math Examples

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

Step 1

Find the common denominator.

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Step 1.1

Multiply $\frac{d - 5}{d^{4}}$ by $\frac{d^{2}}{d^{2}}$ .

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

Step 1.2

Multiply $\frac{d - 5}{d^{4}}$ by $\frac{d^{2}}{d^{2}}$ .

$\frac{1 - 5 d^{2}}{d^{6}} - \frac{(d - 5) d^{2}}{d^{4} d^{2}} + \frac{1}{d^{3}}$

Step 1.3

Multiply $\frac{1}{d^{3}}$ by $\frac{d^{3}}{d^{3}}$ .

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

Step 1.4

Multiply $\frac{1}{d^{3}}$ by $\frac{d^{3}}{d^{3}}$ .

$\frac{1 - 5 d^{2}}{d^{6}} - \frac{(d - 5) d^{2}}{d^{4} d^{2}} + \frac{d^{3}}{d^{3} d^{3}}$

Step 1.5

Multiply $d^{4}$ by $d^{2}$ by adding the exponents.

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Step 1.5.1

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

$\frac{1 - 5 d^{2}}{d^{6}} - \frac{(d - 5) d^{2}}{d^{4 + 2}} + \frac{d^{3}}{d^{3} d^{3}}$

Step 1.5.2

Add $4$ and $2$ .

$\frac{1 - 5 d^{2}}{d^{6}} - \frac{(d - 5) d^{2}}{d^{6}} + \frac{d^{3}}{d^{3} d^{3}}$

Step 1.6

Multiply $d^{3}$ by $d^{3}$ by adding the exponents.

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Step 1.6.1

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

$\frac{1 - 5 d^{2}}{d^{6}} - \frac{(d - 5) d^{2}}{d^{6}} + \frac{d^{3}}{d^{3 + 3}}$

Step 1.6.2

Add $3$ and $3$ .

$\frac{1 - 5 d^{2}}{d^{6}} - \frac{(d - 5) d^{2}}{d^{6}} + \frac{d^{3}}{d^{6}}$

Step 2

Combine the numerators over the common denominator.

$\frac{1 - 5 d^{2} - (d - 5) d^{2} + d^{3}}{d^{6}}$

Step 3

Simplify each term.

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Step 3.1

Apply the distributive property.

$\frac{1 - 5 d^{2} + (- d - - 5) d^{2} + d^{3}}{d^{6}}$

Step 3.2

Multiply $- 1$ by $- 5$ .

$\frac{1 - 5 d^{2} + (- d + 5) d^{2} + d^{3}}{d^{6}}$

Step 3.3

Apply the distributive property.

$\frac{1 - 5 d^{2} - d \cdot d^{2} + 5 d^{2} + d^{3}}{d^{6}}$

Step 3.4

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

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Step 3.4.1

Move $d^{2}$ .

$\frac{1 - 5 d^{2} - (d^{2} d) + 5 d^{2} + d^{3}}{d^{6}}$

Step 3.4.2

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

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Step 3.4.2.1

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

$\frac{1 - 5 d^{2} - (d^{2} d^{1}) + 5 d^{2} + d^{3}}{d^{6}}$

Step 3.4.2.2

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

$\frac{1 - 5 d^{2} - d^{2 + 1} + 5 d^{2} + d^{3}}{d^{6}}$

Step 3.4.3

Add $2$ and $1$ .

$\frac{1 - 5 d^{2} - d^{3} + 5 d^{2} + d^{3}}{d^{6}}$

Step 4

Combine the opposite terms in $1 - 5 d^{2} - d^{3} + 5 d^{2} + d^{3}$ .

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Step 4.1

Add $- 5 d^{2}$ and $5 d^{2}$ .

$\frac{1 - d^{3} + 0 + d^{3}}{d^{6}}$

Step 4.2

Add $1 - d^{3}$ and $0$ .

$\frac{1 - d^{3} + d^{3}}{d^{6}}$

Step 4.3

Add $- d^{3}$ and $d^{3}$ .

$\frac{1 + 0}{d^{6}}$

Step 4.4

Add $1$ and $0$ .

$\frac{1}{d^{6}}$