Algebra Examples

$\frac{{(2 a^{5} b^{3})}^{2} \cdot 12 a b^{4}}{5 a^{7} b^{3} + 3 a^{7} b^{3}}$

Step 1

Simplify the numerator.

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Step 1.1

Apply the product rule to $2 a^{5} b^{3}$ .

$\frac{{(2 a^{5})}^{2} {(b^{3})}^{2} \cdot 12 a b^{4}}{5 a^{7} b^{3} + 3 a^{7} b^{3}}$

Step 1.2

Apply the product rule to $2 a^{5}$ .

$\frac{2^{2} {(a^{5})}^{2} {(b^{3})}^{2} \cdot 12 a b^{4}}{5 a^{7} b^{3} + 3 a^{7} b^{3}}$

Step 1.3

Raise $2$ to the power of $2$ .

$\frac{4 {(a^{5})}^{2} {(b^{3})}^{2} \cdot 12 a b^{4}}{5 a^{7} b^{3} + 3 a^{7} b^{3}}$

Step 1.4

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

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Step 1.4.1

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

$\frac{4 a^{5 \cdot 2} {(b^{3})}^{2} \cdot 12 a b^{4}}{5 a^{7} b^{3} + 3 a^{7} b^{3}}$

Step 1.4.2

Multiply $5$ by $2$ .

$\frac{4 a^{10} {(b^{3})}^{2} \cdot 12 a b^{4}}{5 a^{7} b^{3} + 3 a^{7} b^{3}}$

Step 1.5

Multiply the exponents in ${(b^{3})}^{2}$ .

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Step 1.5.1

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

$\frac{4 a^{10} b^{3 \cdot 2} \cdot 12 a b^{4}}{5 a^{7} b^{3} + 3 a^{7} b^{3}}$

Step 1.5.2

Multiply $3$ by $2$ .

$\frac{4 a^{10} b^{6} \cdot 12 a b^{4}}{5 a^{7} b^{3} + 3 a^{7} b^{3}}$

Step 1.6

Combine exponents.

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Step 1.6.1

Multiply $12$ by $4$ .

$\frac{48 a^{10} b^{6} a b^{4}}{5 a^{7} b^{3} + 3 a^{7} b^{3}}$

Step 1.6.2

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

$\frac{48 (a^{1} a^{10}) b^{6} b^{4}}{5 a^{7} b^{3} + 3 a^{7} b^{3}}$

Step 1.6.3

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

$\frac{48 a^{1 + 10} b^{6} b^{4}}{5 a^{7} b^{3} + 3 a^{7} b^{3}}$

Step 1.6.4

Add $1$ and $10$ .

$\frac{48 a^{11} b^{6} b^{4}}{5 a^{7} b^{3} + 3 a^{7} b^{3}}$

Step 1.6.5

Multiply $b^{6}$ by $b^{4}$ by adding the exponents.

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Step 1.6.5.1

Move $b^{4}$ .

$\frac{48 a^{11} (b^{4} b^{6})}{5 a^{7} b^{3} + 3 a^{7} b^{3}}$

Step 1.6.5.2

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

$\frac{48 a^{11} b^{4 + 6}}{5 a^{7} b^{3} + 3 a^{7} b^{3}}$

Step 1.6.5.3

Add $4$ and $6$ .

$\frac{48 a^{11} b^{10}}{5 a^{7} b^{3} + 3 a^{7} b^{3}}$

Step 2

Simplify terms.

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Step 2.1

Add $5 a^{7} b^{3}$ and $3 a^{7} b^{3}$ .

$\frac{48 a^{11} b^{10}}{8 a^{7} b^{3}}$

Step 2.2

Cancel the common factor of $48$ and $8$ .

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Step 2.2.1

Factor $8$ out of $48 a^{11} b^{10}$ .

$\frac{8 (6 a^{11} b^{10})}{8 a^{7} b^{3}}$

Step 2.2.2

Cancel the common factors.

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Step 2.2.2.1

Factor $8$ out of $8 a^{7} b^{3}$ .

$\frac{8 (6 a^{11} b^{10})}{8 (a^{7} b^{3})}$

Step 2.2.2.2

Cancel the common factor.

$\frac{8 (6 a^{11} b^{10})}{8 (a^{7} b^{3})}$

Step 2.2.2.3

Rewrite the expression.

$\frac{6 a^{11} b^{10}}{a^{7} b^{3}}$

Step 2.3

Cancel the common factor of $a^{11}$ and $a^{7}$ .

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Step 2.3.1

Factor $a^{7}$ out of $6 a^{11} b^{10}$ .

$\frac{a^{7} (6 a^{4} b^{10})}{a^{7} b^{3}}$

Step 2.3.2

Cancel the common factors.

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Step 2.3.2.1

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

$\frac{a^{7} (6 a^{4} b^{10})}{a^{7} (b^{3})}$

Step 2.3.2.2

Cancel the common factor.

$\frac{a^{7} (6 a^{4} b^{10})}{a^{7} b^{3}}$

Step 2.3.2.3

Rewrite the expression.

$\frac{6 a^{4} b^{10}}{b^{3}}$

Step 2.4

Cancel the common factor of $b^{10}$ and $b^{3}$ .

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Step 2.4.1

Factor $b^{3}$ out of $6 a^{4} b^{10}$ .

$\frac{b^{3} (6 a^{4} b^{7})}{b^{3}}$

Step 2.4.2

Cancel the common factors.

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Step 2.4.2.1

Multiply by $1$ .

$\frac{b^{3} (6 a^{4} b^{7})}{b^{3} \cdot 1}$

Step 2.4.2.2

Cancel the common factor.

$\frac{b^{3} (6 a^{4} b^{7})}{b^{3} \cdot 1}$

Step 2.4.2.3

Rewrite the expression.

$\frac{6 a^{4} b^{7}}{1}$

Step 2.4.2.4

Divide $6 a^{4} b^{7}$ by $1$ .

$6 a^{4} b^{7}$