Pre-Algebra Examples

$\frac{8}{9 u^{3} v^{2}} + \frac{8}{30 u^{5} v^{5}}$

Step 1

Cancel the common factor of $8$ and $30$ .

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Step 1.1

Factor $2$ out of $8$ .

$\frac{8}{9 u^{3} v^{2}} + \frac{2 (4)}{30 u^{5} v^{5}}$

Step 1.2

Cancel the common factors.

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Step 1.2.1

Factor $2$ out of $30 u^{5} v^{5}$ .

$\frac{8}{9 u^{3} v^{2}} + \frac{2 (4)}{2 (15 u^{5} v^{5})}$

Step 1.2.2

Cancel the common factor.

$\frac{8}{9 u^{3} v^{2}} + \frac{2 \cdot 4}{2 (15 u^{5} v^{5})}$

Step 1.2.3

Rewrite the expression.

$\frac{8}{9 u^{3} v^{2}} + \frac{4}{15 u^{5} v^{5}}$

Step 2

To write $\frac{8}{9 u^{3} v^{2}}$ as a fraction with a common denominator, multiply by $\frac{5 u^{2} v^{3}}{5 u^{2} v^{3}}$ .

$\frac{8}{9 u^{3} v^{2}} \cdot \frac{5 u^{2} v^{3}}{5 u^{2} v^{3}} + \frac{4}{15 u^{5} v^{5}}$

Step 3

To write $\frac{4}{15 u^{5} v^{5}}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{8}{9 u^{3} v^{2}} \cdot \frac{5 u^{2} v^{3}}{5 u^{2} v^{3}} + \frac{4}{15 u^{5} v^{5}} \cdot \frac{3}{3}$

Step 4

Write each expression with a common denominator of $45 u^{5} v^{5}$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{8}{9 u^{3} v^{2}}$ by $\frac{5 u^{2} v^{3}}{5 u^{2} v^{3}}$ .

$\frac{8 (5 u^{2} v^{3})}{9 u^{3} v^{2} (5 u^{2} v^{3})} + \frac{4}{15 u^{5} v^{5}} \cdot \frac{3}{3}$

Step 4.2

Multiply $5$ by $9$ .

$\frac{8 (5 u^{2} v^{3})}{45 u^{3} v^{2} (u^{2} v^{3})} + \frac{4}{15 u^{5} v^{5}} \cdot \frac{3}{3}$

Step 4.3

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

$\frac{8 (5 u^{2} v^{3})}{45 u^{2 + 3} v^{2} v^{3}} + \frac{4}{15 u^{5} v^{5}} \cdot \frac{3}{3}$

Step 4.4

Add $2$ and $3$ .

$\frac{8 (5 u^{2} v^{3})}{45 u^{5} v^{2} v^{3}} + \frac{4}{15 u^{5} v^{5}} \cdot \frac{3}{3}$

Step 4.5

Multiply $v^{2}$ by $v^{3}$ by adding the exponents.

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Step 4.5.1

Move $v^{3}$ .

$\frac{8 (5 u^{2} v^{3})}{45 u^{5} (v^{3} v^{2})} + \frac{4}{15 u^{5} v^{5}} \cdot \frac{3}{3}$

Step 4.5.2

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

$\frac{8 (5 u^{2} v^{3})}{45 u^{5} v^{3 + 2}} + \frac{4}{15 u^{5} v^{5}} \cdot \frac{3}{3}$

Step 4.5.3

Add $3$ and $2$ .

$\frac{8 (5 u^{2} v^{3})}{45 u^{5} v^{5}} + \frac{4}{15 u^{5} v^{5}} \cdot \frac{3}{3}$

Step 4.6

Multiply $\frac{4}{15 u^{5} v^{5}}$ by $\frac{3}{3}$ .

$\frac{8 (5 u^{2} v^{3})}{45 u^{5} v^{5}} + \frac{4 \cdot 3}{15 u^{5} v^{5} \cdot 3}$

Step 4.7

Multiply $3$ by $15$ .

$\frac{8 (5 u^{2} v^{3})}{45 u^{5} v^{5}} + \frac{4 \cdot 3}{45 u^{5} v^{5}}$

Step 5

Combine the numerators over the common denominator.

$\frac{8 (5 u^{2} v^{3}) + 4 \cdot 3}{45 u^{5} v^{5}}$

Step 6

Simplify the numerator.

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Step 6.1

Factor $4$ out of $8 \cdot 5 u^{2} v^{3} + 4 \cdot 3$ .

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Step 6.1.1

Factor $4$ out of $8 \cdot 5 u^{2} v^{3}$ .

$\frac{4 (2 \cdot 5 u^{2} v^{3}) + 4 \cdot 3}{45 u^{5} v^{5}}$

Step 6.1.2

Factor $4$ out of $4 \cdot 3$ .

$\frac{4 (2 \cdot 5 u^{2} v^{3}) + 4 (3)}{45 u^{5} v^{5}}$

Step 6.1.3

Factor $4$ out of $4 (2 \cdot 5 u^{2} v^{3}) + 4 (3)$ .

$\frac{4 (2 \cdot 5 u^{2} v^{3} + 3)}{45 u^{5} v^{5}}$

Step 6.2

Multiply $2$ by $5$ .

$\frac{4 (10 u^{2} v^{3} + 3)}{45 u^{5} v^{5}}$