Algebra Examples

$\frac{40 v^{2}}{35 v^{4}} + \frac{20 v^{3}}{5 v}$

Step 1

Simplify each term.

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Step 1.1

Cancel the common factor of $40$ and $35$ .

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Step 1.1.1

Factor $5$ out of $40 v^{2}$ .

$\frac{5 (8 v^{2})}{35 v^{4}} + \frac{20 v^{3}}{5 v}$

Step 1.1.2

Cancel the common factors.

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Step 1.1.2.1

Factor $5$ out of $35 v^{4}$ .

$\frac{5 (8 v^{2})}{5 (7 v^{4})} + \frac{20 v^{3}}{5 v}$

Step 1.1.2.2

Cancel the common factor.

$\frac{5 (8 v^{2})}{5 (7 v^{4})} + \frac{20 v^{3}}{5 v}$

Step 1.1.2.3

Rewrite the expression.

$\frac{8 v^{2}}{7 v^{4}} + \frac{20 v^{3}}{5 v}$

Step 1.2

Cancel the common factor of $v^{2}$ and $v^{4}$ .

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Step 1.2.1

Factor $v^{2}$ out of $8 v^{2}$ .

$\frac{v^{2} \cdot 8}{7 v^{4}} + \frac{20 v^{3}}{5 v}$

Step 1.2.2

Cancel the common factors.

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Step 1.2.2.1

Factor $v^{2}$ out of $7 v^{4}$ .

$\frac{v^{2} \cdot 8}{v^{2} (7 v^{2})} + \frac{20 v^{3}}{5 v}$

Step 1.2.2.2

Cancel the common factor.

$\frac{v^{2} \cdot 8}{v^{2} (7 v^{2})} + \frac{20 v^{3}}{5 v}$

Step 1.2.2.3

Rewrite the expression.

$\frac{8}{7 v^{2}} + \frac{20 v^{3}}{5 v}$

Step 1.3

Cancel the common factor of $20$ and $5$ .

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Step 1.3.1

Factor $5$ out of $20 v^{3}$ .

$\frac{8}{7 v^{2}} + \frac{5 (4 v^{3})}{5 v}$

Step 1.3.2

Cancel the common factors.

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Step 1.3.2.1

Factor $5$ out of $5 v$ .

$\frac{8}{7 v^{2}} + \frac{5 (4 v^{3})}{5 (v)}$

Step 1.3.2.2

Cancel the common factor.

$\frac{8}{7 v^{2}} + \frac{5 (4 v^{3})}{5 v}$

Step 1.3.2.3

Rewrite the expression.

$\frac{8}{7 v^{2}} + \frac{4 v^{3}}{v}$

Step 1.4

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

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Step 1.4.1

Factor $v$ out of $4 v^{3}$ .

$\frac{8}{7 v^{2}} + \frac{v (4 v^{2})}{v}$

Step 1.4.2

Cancel the common factors.

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Step 1.4.2.1

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

$\frac{8}{7 v^{2}} + \frac{v (4 v^{2})}{v^{1}}$

Step 1.4.2.2

Factor $v$ out of $v^{1}$ .

$\frac{8}{7 v^{2}} + \frac{v (4 v^{2})}{v \cdot 1}$

Step 1.4.2.3

Cancel the common factor.

$\frac{8}{7 v^{2}} + \frac{v (4 v^{2})}{v \cdot 1}$

Step 1.4.2.4

Rewrite the expression.

$\frac{8}{7 v^{2}} + \frac{4 v^{2}}{1}$

Step 1.4.2.5

Divide $4 v^{2}$ by $1$ .

$\frac{8}{7 v^{2}} + 4 v^{2}$

Step 2

To write $4 v^{2}$ as a fraction with a common denominator, multiply by $\frac{7 v^{2}}{7 v^{2}}$ .

$\frac{8}{7 v^{2}} + 4 v^{2} \cdot \frac{7 v^{2}}{7 v^{2}}$

Step 3

Simplify terms.

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Step 3.1

Combine $4 v^{2}$ and $\frac{7 v^{2}}{7 v^{2}}$ .

$\frac{8}{7 v^{2}} + \frac{4 v^{2} (7 v^{2})}{7 v^{2}}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{8 + 4 v^{2} (7 v^{2})}{7 v^{2}}$

Step 4

Simplify the numerator.

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Step 4.1

Rewrite using the commutative property of multiplication.

$\frac{8 + 4 \cdot 7 v^{2} v^{2}}{7 v^{2}}$

Step 4.2

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

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Step 4.2.1

Move $v^{2}$ .

$\frac{8 + 4 \cdot 7 (v^{2} v^{2})}{7 v^{2}}$

Step 4.2.2

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

$\frac{8 + 4 \cdot 7 v^{2 + 2}}{7 v^{2}}$

Step 4.2.3

Add $2$ and $2$ .

$\frac{8 + 4 \cdot 7 v^{4}}{7 v^{2}}$

Step 4.3

Multiply $4$ by $7$ .

$\frac{8 + 28 v^{4}}{7 v^{2}}$

Step 4.4

Factor $4$ out of $8 + 28 v^{4}$ .

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Step 4.4.1

Factor $4$ out of $8$ .

$\frac{4 (2) + 28 v^{4}}{7 v^{2}}$

Step 4.4.2

Factor $4$ out of $28 v^{4}$ .

$\frac{4 (2) + 4 (7 v^{4})}{7 v^{2}}$

Step 4.4.3

Factor $4$ out of $4 (2) + 4 (7 v^{4})$ .

$\frac{4 (2 + 7 v^{4})}{7 v^{2}}$