Basic Math Examples

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

Step 1

To write $\frac{5 a - 1}{2 a^{4}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 2

To write $\frac{3 a + 1}{4 a^{2}}$ as a fraction with a common denominator, multiply by $\frac{a^{2}}{a^{2}}$ .

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

Step 3

Write each expression with a common denominator of $4 a^{4}$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{5 a - 1}{2 a^{4}}$ by $\frac{2}{2}$ .

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

Step 3.2

Multiply $2$ by $2$ .

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

Step 3.3

Multiply $\frac{3 a + 1}{4 a^{2}}$ by $\frac{a^{2}}{a^{2}}$ .

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

Step 3.4

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

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Step 3.4.1

Move $a^{2}$ .

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

Step 3.4.2

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

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

Step 3.4.3

Add $2$ and $2$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Multiply $2$ by $5$ .

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

Step 5.3

Multiply $- 1$ by $2$ .

$\frac{10 a - 2 + (3 a + 1) a^{2}}{4 a^{4}}$

Step 5.4

Apply the distributive property.

$\frac{10 a - 2 + 3 a \cdot a^{2} + 1 a^{2}}{4 a^{4}}$

Step 5.5

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

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Step 5.5.1

Move $a^{2}$ .

$\frac{10 a - 2 + 3 (a^{2} a) + 1 a^{2}}{4 a^{4}}$

Step 5.5.2

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

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Step 5.5.2.1

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

$\frac{10 a - 2 + 3 (a^{2} a^{1}) + 1 a^{2}}{4 a^{4}}$

Step 5.5.2.2

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

$\frac{10 a - 2 + 3 a^{2 + 1} + 1 a^{2}}{4 a^{4}}$

Step 5.5.3

Add $2$ and $1$ .

$\frac{10 a - 2 + 3 a^{3} + 1 a^{2}}{4 a^{4}}$

Step 5.6

Multiply $a^{2}$ by $1$ .

$\frac{10 a - 2 + 3 a^{3} + a^{2}}{4 a^{4}}$

Step 5.7

Reorder terms.

$\frac{3 a^{3} + a^{2} + 10 a - 2}{4 a^{4}}$