Algebra Examples

$\frac{10 a^{5} b^{2}}{2 b^{7}} \cdot \frac{4 {(a^{2})}^{5}}{b^{- 4}}$

Step 1

Combine.

$\frac{10 a^{5} b^{2} (4 {(a^{2})}^{5})}{2 b^{7} b^{- 4}}$

Step 2

Multiply $b^{7}$ by $b^{- 4}$ by adding the exponents.

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Step 2.1

Move $b^{- 4}$ .

$\frac{10 a^{5} b^{2} (4 {(a^{2})}^{5})}{2 (b^{- 4} b^{7})}$

Step 2.2

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

$\frac{10 a^{5} b^{2} (4 {(a^{2})}^{5})}{2 b^{- 4 + 7}}$

Step 2.3

Add $- 4$ and $7$ .

$\frac{10 a^{5} b^{2} (4 {(a^{2})}^{5})}{2 b^{3}}$

Step 3

Reduce the expression by cancelling the common factors.

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Step 3.1

Cancel the common factor of $10$ and $2$ .

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Step 3.1.1

Factor $2$ out of $10 a^{5} b^{2} (4 {(a^{2})}^{5})$ .

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

Step 3.1.2

Cancel the common factors.

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Step 3.1.2.1

Factor $2$ out of $2 b^{3}$ .

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

Step 3.1.2.2

Cancel the common factor.

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

Step 3.1.2.3

Rewrite the expression.

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

Step 3.2

Cancel the common factor of $b^{2}$ and $b^{3}$ .

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Step 3.2.1

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

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

Step 3.2.2

Cancel the common factors.

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Step 3.2.2.1

Factor $b^{2}$ out of $b^{3}$ .

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

Step 3.2.2.2

Cancel the common factor.

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

Step 3.2.2.3

Rewrite the expression.

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

Step 3.3

Multiply $4$ by $5$ .

$\frac{20 a^{5} {(a^{2})}^{5}}{b}$

Step 4

Simplify the numerator.

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Step 4.1

Multiply the exponents in ${(a^{2})}^{5}$ .

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Step 4.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{20 a^{5} a^{2 \cdot 5}}{b}$

Step 4.1.2

Multiply $2$ by $5$ .

$\frac{20 a^{5} a^{10}}{b}$

Step 4.2

Multiply $a^{5}$ by $a^{10}$ by adding the exponents.

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Step 4.2.1

Move $a^{10}$ .

$\frac{20 (a^{10} a^{5})}{b}$

Step 4.2.2

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

$\frac{20 a^{10 + 5}}{b}$

Step 4.2.3

Add $10$ and $5$ .

$\frac{20 a^{15}}{b}$