Basic Math Examples

${(a^{2} b^{3})}^{5} {(2 a^{3} b^{5})}^{2} \div 4 a^{7} b^{9}$

Step 1

Rewrite the division as a fraction.

$\frac{{(a^{2} b^{3})}^{5} {(2 a^{3} b^{5})}^{2}}{4 a^{7} b^{9}}$

Step 2

Simplify the numerator.

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Step 2.1

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

$\frac{{(a^{2})}^{5} {(b^{3})}^{5} {(2 a^{3} b^{5})}^{2}}{4 a^{7} b^{9}}$

Step 2.2

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

$\frac{{(a^{2})}^{5} {(b^{3})}^{5} {(2 a^{3})}^{2} {(b^{5})}^{2}}{4 a^{7} b^{9}}$

Step 2.3

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

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Step 2.3.1

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

$\frac{a^{2 \cdot 5} {(b^{3})}^{5} {(2 a^{3})}^{2} {(b^{5})}^{2}}{4 a^{7} b^{9}}$

Step 2.3.2

Multiply $2$ by $5$ .

$\frac{a^{10} {(b^{3})}^{5} {(2 a^{3})}^{2} {(b^{5})}^{2}}{4 a^{7} b^{9}}$

Step 2.4

Multiply the exponents in ${(b^{3})}^{5}$ .

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Step 2.4.1

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

$\frac{a^{10} b^{3 \cdot 5} {(2 a^{3})}^{2} {(b^{5})}^{2}}{4 a^{7} b^{9}}$

Step 2.4.2

Multiply $3$ by $5$ .

$\frac{a^{10} b^{15} {(2 a^{3})}^{2} {(b^{5})}^{2}}{4 a^{7} b^{9}}$

Step 2.5

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

$\frac{a^{10} b^{15} (2^{2} {(a^{3})}^{2}) {(b^{5})}^{2}}{4 a^{7} b^{9}}$

Step 2.6

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

$\frac{a^{10} b^{15} (4 {(a^{3})}^{2}) {(b^{5})}^{2}}{4 a^{7} b^{9}}$

Step 2.7

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

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Step 2.7.1

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

$\frac{a^{10} b^{15} (4 a^{3 \cdot 2}) {(b^{5})}^{2}}{4 a^{7} b^{9}}$

Step 2.7.2

Multiply $3$ by $2$ .

$\frac{a^{10} b^{15} (4 a^{6}) {(b^{5})}^{2}}{4 a^{7} b^{9}}$

Step 2.8

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

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Step 2.8.1

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

$\frac{a^{10} b^{15} (4 a^{6}) b^{5 \cdot 2}}{4 a^{7} b^{9}}$

Step 2.8.2

Multiply $5$ by $2$ .

$\frac{a^{10} b^{15} (4 a^{6}) b^{10}}{4 a^{7} b^{9}}$

$\frac{a^{10} b^{15} \cdot 4 a^{6} b^{10}}{4 a^{7} b^{9}}$

Step 2.9

Combine exponents.

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Step 2.9.1

Multiply $a^{10}$ by $a^{6}$ by adding the exponents.

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Step 2.9.1.1

Move $a^{6}$ .

$\frac{a^{6} a^{10} b^{15} \cdot 4 b^{10}}{4 a^{7} b^{9}}$

Step 2.9.1.2

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

$\frac{a^{6 + 10} b^{15} \cdot 4 b^{10}}{4 a^{7} b^{9}}$

Step 2.9.1.3

Add $6$ and $10$ .

$\frac{a^{16} b^{15} \cdot 4 b^{10}}{4 a^{7} b^{9}}$

Step 2.9.2

Multiply $b^{15}$ by $b^{10}$ by adding the exponents.

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Step 2.9.2.1

Move $b^{10}$ .

$\frac{a^{16} (b^{10} b^{15}) \cdot 4}{4 a^{7} b^{9}}$

Step 2.9.2.2

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

$\frac{a^{16} b^{10 + 15} \cdot 4}{4 a^{7} b^{9}}$

Step 2.9.2.3

Add $10$ and $15$ .

$\frac{a^{16} b^{25} \cdot 4}{4 a^{7} b^{9}}$

Step 3

Reduce the expression by cancelling the common factors.

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Step 3.1

Cancel the common factor of $a^{16}$ and $a^{7}$ .

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Step 3.1.1

Factor $a^{7}$ out of $a^{16} b^{25} \cdot 4$ .

$\frac{a^{7} (a^{9} b^{25} \cdot 4)}{4 a^{7} b^{9}}$

Step 3.1.2

Cancel the common factors.

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Step 3.1.2.1

Factor $a^{7}$ out of $4 a^{7} b^{9}$ .

$\frac{a^{7} (a^{9} b^{25} \cdot 4)}{a^{7} (4 b^{9})}$

Step 3.1.2.2

Cancel the common factor.

$\frac{a^{7} (a^{9} b^{25} \cdot 4)}{a^{7} (4 b^{9})}$

Step 3.1.2.3

Rewrite the expression.

$\frac{a^{9} b^{25} \cdot 4}{4 b^{9}}$

Step 3.2

Cancel the common factor of $b^{25}$ and $b^{9}$ .

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Step 3.2.1

Factor $b^{9}$ out of $a^{9} b^{25} \cdot 4$ .

$\frac{b^{9} (a^{9} b^{16} \cdot 4)}{4 b^{9}}$

Step 3.2.2

Cancel the common factors.

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Step 3.2.2.1

Factor $b^{9}$ out of $4 b^{9}$ .

$\frac{b^{9} (a^{9} b^{16} \cdot 4)}{b^{9} \cdot 4}$

Step 3.2.2.2

Cancel the common factor.

$\frac{b^{9} (a^{9} b^{16} \cdot 4)}{b^{9} \cdot 4}$

Step 3.2.2.3

Rewrite the expression.

$\frac{a^{9} b^{16} \cdot 4}{4}$

Step 3.3

Cancel the common factor of $4$ .

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Step 3.3.1

Cancel the common factor.

$\frac{a^{9} b^{16} \cdot 4}{4}$

Step 3.3.2

Divide $a^{9} b^{16}$ by $1$ .

$a^{9} b^{16}$