Basic Math Examples

${(3 p^{2} q)}^{2} \cdot {(q^{2})}^{3} \div p^{6} q^{11}$

Step 1

Rewrite the division as a fraction.

$\frac{{(3 p^{2} q)}^{2} \cdot {(q^{2})}^{3}}{p^{6} q^{11}}$

Step 2

Simplify the numerator.

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Step 2.1

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

$\frac{{(3 p^{2})}^{2} q^{2} {(q^{2})}^{3}}{p^{6} q^{11}}$

Step 2.2

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

$\frac{3^{2} {(p^{2})}^{2} q^{2} {(q^{2})}^{3}}{p^{6} q^{11}}$

Step 2.3

Raise $3$ to the power of $2$ .

$\frac{9 {(p^{2})}^{2} q^{2} {(q^{2})}^{3}}{p^{6} q^{11}}$

Step 2.4

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

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Step 2.4.1

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

$\frac{9 p^{2 \cdot 2} q^{2} {(q^{2})}^{3}}{p^{6} q^{11}}$

Step 2.4.2

Multiply $2$ by $2$ .

$\frac{9 p^{4} q^{2} {(q^{2})}^{3}}{p^{6} q^{11}}$

Step 2.5

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

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Step 2.5.1

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

$\frac{9 p^{4} q^{2} q^{2 \cdot 3}}{p^{6} q^{11}}$

Step 2.5.2

Multiply $2$ by $3$ .

$\frac{9 p^{4} q^{2} q^{6}}{p^{6} q^{11}}$

Step 2.6

Multiply $q^{2}$ by $q^{6}$ by adding the exponents.

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Step 2.6.1

Move $q^{6}$ .

$\frac{9 p^{4} (q^{6} q^{2})}{p^{6} q^{11}}$

Step 2.6.2

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

$\frac{9 p^{4} q^{6 + 2}}{p^{6} q^{11}}$

Step 2.6.3

Add $6$ and $2$ .

$\frac{9 p^{4} q^{8}}{p^{6} q^{11}}$

Step 3

Reduce the expression by cancelling the common factors.

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Step 3.1

Cancel the common factor of $p^{4}$ and $p^{6}$ .

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Step 3.1.1

Factor $p^{4}$ out of $9 p^{4} q^{8}$ .

$\frac{p^{4} (9 q^{8})}{p^{6} q^{11}}$

Step 3.1.2

Cancel the common factors.

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Step 3.1.2.1

Factor $p^{4}$ out of $p^{6} q^{11}$ .

$\frac{p^{4} (9 q^{8})}{p^{4} (p^{2} q^{11})}$

Step 3.1.2.2

Cancel the common factor.

$\frac{p^{4} (9 q^{8})}{p^{4} (p^{2} q^{11})}$

Step 3.1.2.3

Rewrite the expression.

$\frac{9 q^{8}}{p^{2} q^{11}}$

Step 3.2

Cancel the common factor of $q^{8}$ and $q^{11}$ .

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Step 3.2.1

Factor $q^{8}$ out of $9 q^{8}$ .

$\frac{q^{8} \cdot 9}{p^{2} q^{11}}$

Step 3.2.2

Cancel the common factors.

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Step 3.2.2.1

Factor $q^{8}$ out of $p^{2} q^{11}$ .

$\frac{q^{8} \cdot 9}{q^{8} (p^{2} q^{3})}$

Step 3.2.2.2

Cancel the common factor.

$\frac{q^{8} \cdot 9}{q^{8} (p^{2} q^{3})}$

Step 3.2.2.3

Rewrite the expression.

$\frac{9}{p^{2} q^{3}}$