Basic Math Examples

$\frac{a^{4}}{b^{4}} \cdot (\frac{3 b^{5}}{a^{4}} + \frac{5 b^{7}}{a^{2}})$

Step 1

To write $\frac{5 b^{7}}{a^{2}}$ as a fraction with a common denominator, multiply by $\frac{a^{2}}{a^{2}}$ .

$\frac{a^{4}}{b^{4}} \cdot (\frac{3 b^{5}}{a^{4}} + \frac{5 b^{7}}{a^{2}} \cdot \frac{a^{2}}{a^{2}})$

Step 2

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

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Step 2.1

Multiply $\frac{5 b^{7}}{a^{2}}$ by $\frac{a^{2}}{a^{2}}$ .

$\frac{a^{4}}{b^{4}} \cdot (\frac{3 b^{5}}{a^{4}} + \frac{5 b^{7} a^{2}}{a^{2} a^{2}})$

Step 2.2

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

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Step 2.2.1

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

$\frac{a^{4}}{b^{4}} \cdot (\frac{3 b^{5}}{a^{4}} + \frac{5 b^{7} a^{2}}{a^{2 + 2}})$

Step 2.2.2

Add $2$ and $2$ .

$\frac{a^{4}}{b^{4}} \cdot (\frac{3 b^{5}}{a^{4}} + \frac{5 b^{7} a^{2}}{a^{4}})$

Step 3

Simplify terms.

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Step 3.1

Combine the numerators over the common denominator.

$\frac{a^{4}}{b^{4}} \cdot \frac{3 b^{5} + 5 b^{7} a^{2}}{a^{4}}$

Step 3.2

Factor $b^{5}$ out of $3 b^{5} + 5 b^{7} a^{2}$ .

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Step 3.2.1

Factor $b^{5}$ out of $3 b^{5}$ .

$\frac{a^{4}}{b^{4}} \cdot \frac{b^{5} \cdot 3 + 5 b^{7} a^{2}}{a^{4}}$

Step 3.2.2

Factor $b^{5}$ out of $5 b^{7} a^{2}$ .

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

Step 3.2.3

Factor $b^{5}$ out of $b^{5} \cdot 3 + b^{5} (5 b^{2} a^{2})$ .

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

Step 3.3

Combine.

$\frac{a^{4} (b^{5} (3 + 5 b^{2} a^{2}))}{b^{4} a^{4}}$

Step 3.4

Cancel the common factor of $a^{4}$ .

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Step 3.4.1

Cancel the common factor.

$\frac{a^{4} (b^{5} (3 + 5 b^{2} a^{2}))}{b^{4} a^{4}}$

Step 3.4.2

Rewrite the expression.

$\frac{b^{5} (3 + 5 b^{2} a^{2})}{b^{4}}$

Step 3.5

Cancel the common factor of $b^{5}$ and $b^{4}$ .

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Step 3.5.1

Factor $b^{4}$ out of $b^{5} (3 + 5 b^{2} a^{2})$ .

$\frac{b^{4} (b (3 + 5 b^{2} a^{2}))}{b^{4}}$

Step 3.5.2

Cancel the common factors.

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Step 3.5.2.1

Multiply by $1$ .

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

Step 3.5.2.2

Cancel the common factor.

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

Step 3.5.2.3

Rewrite the expression.

$\frac{b (3 + 5 b^{2} a^{2})}{1}$

Step 3.5.2.4

Divide $b (3 + 5 b^{2} a^{2})$ by $1$ .

$b (3 + 5 b^{2} a^{2})$

Step 3.6

Apply the distributive property.

$b \cdot 3 + b (5 b^{2} a^{2})$

Step 3.7

Reorder.

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Step 3.7.1

Move $3$ to the left of $b$ .

$3 \cdot b + b (5 b^{2} a^{2})$

Step 3.7.2

Rewrite using the commutative property of multiplication.

$3 \cdot b + 5 b (b^{2} a^{2})$

Step 4

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

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Step 4.1

Move $b^{2}$ .

$3 b + 5 (b^{2} b) a^{2}$

Step 4.2

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

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Step 4.2.1

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

$3 b + 5 (b^{2} b^{1}) a^{2}$

Step 4.2.2

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

$3 b + 5 b^{2 + 1} a^{2}$

Step 4.3

Add $2$ and $1$ .

$3 b + 5 b^{3} a^{2}$