Algebra Examples

${(\frac{2 p}{3 q^{2} r^{3}})}^{- 3} \cdot {(\frac{p^{3} r^{2}}{q^{4}})}^{2}$

Step 1

Change the sign of the exponent by rewriting the base as its reciprocal.

${(\frac{3 q^{2} r^{3}}{2 p})}^{3} \cdot {(\frac{p^{3} r^{2}}{q^{4}})}^{2}$

Step 2

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.1

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

$\frac{{(3 q^{2} r^{3})}^{3}}{{(2 p)}^{3}} \cdot {(\frac{p^{3} r^{2}}{q^{4}})}^{2}$

Step 2.2

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

$\frac{{(3 q^{2})}^{3} {(r^{3})}^{3}}{{(2 p)}^{3}} \cdot {(\frac{p^{3} r^{2}}{q^{4}})}^{2}$

Step 2.3

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

$\frac{3^{3} {(q^{2})}^{3} {(r^{3})}^{3}}{{(2 p)}^{3}} \cdot {(\frac{p^{3} r^{2}}{q^{4}})}^{2}$

Step 2.4

Apply the product rule to $2 p$ .

$\frac{3^{3} {(q^{2})}^{3} {(r^{3})}^{3}}{2^{3} p^{3}} \cdot {(\frac{p^{3} r^{2}}{q^{4}})}^{2}$

Step 3

Simplify the numerator.

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Step 3.1

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

$\frac{27 {(q^{2})}^{3} {(r^{3})}^{3}}{2^{3} p^{3}} \cdot {(\frac{p^{3} r^{2}}{q^{4}})}^{2}$

Step 3.2

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

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Step 3.2.1

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

$\frac{27 q^{2 \cdot 3} {(r^{3})}^{3}}{2^{3} p^{3}} \cdot {(\frac{p^{3} r^{2}}{q^{4}})}^{2}$

Step 3.2.2

Multiply $2$ by $3$ .

$\frac{27 q^{6} {(r^{3})}^{3}}{2^{3} p^{3}} \cdot {(\frac{p^{3} r^{2}}{q^{4}})}^{2}$

Step 3.3

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

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Step 3.3.1

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

$\frac{27 q^{6} r^{3 \cdot 3}}{2^{3} p^{3}} \cdot {(\frac{p^{3} r^{2}}{q^{4}})}^{2}$

Step 3.3.2

Multiply $3$ by $3$ .

$\frac{27 q^{6} r^{9}}{2^{3} p^{3}} \cdot {(\frac{p^{3} r^{2}}{q^{4}})}^{2}$

Step 4

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

$\frac{27 q^{6} r^{9}}{8 p^{3}} \cdot {(\frac{p^{3} r^{2}}{q^{4}})}^{2}$

Step 5

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 5.1

Apply the product rule to $\frac{p^{3} r^{2}}{q^{4}}$ .

$\frac{27 q^{6} r^{9}}{8 p^{3}} \cdot \frac{{(p^{3} r^{2})}^{2}}{{(q^{4})}^{2}}$

Step 5.2

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

$\frac{27 q^{6} r^{9}}{8 p^{3}} \cdot \frac{{(p^{3})}^{2} {(r^{2})}^{2}}{{(q^{4})}^{2}}$

Step 6

Combine.

$\frac{27 q^{6} r^{9} ({(p^{3})}^{2} {(r^{2})}^{2})}{8 p^{3} {(q^{4})}^{2}}$

Step 7

Simplify the numerator.

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Step 7.1

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

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Step 7.1.1

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

$\frac{27 q^{6} r^{9} p^{3 \cdot 2} {(r^{2})}^{2}}{8 p^{3} {(q^{4})}^{2}}$

Step 7.1.2

Multiply $3$ by $2$ .

$\frac{27 q^{6} r^{9} p^{6} {(r^{2})}^{2}}{8 p^{3} {(q^{4})}^{2}}$

Step 7.2

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

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Step 7.2.1

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

$\frac{27 q^{6} r^{9} p^{6} r^{2 \cdot 2}}{8 p^{3} {(q^{4})}^{2}}$

Step 7.2.2

Multiply $2$ by $2$ .

$\frac{27 q^{6} r^{9} p^{6} r^{4}}{8 p^{3} {(q^{4})}^{2}}$

Step 7.3

Multiply $r^{9}$ by $r^{4}$ by adding the exponents.

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Step 7.3.1

Move $r^{4}$ .

$\frac{27 q^{6} (r^{4} r^{9}) p^{6}}{8 p^{3} {(q^{4})}^{2}}$

Step 7.3.2

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

$\frac{27 q^{6} r^{4 + 9} p^{6}}{8 p^{3} {(q^{4})}^{2}}$

Step 7.3.3

Add $4$ and $9$ .

$\frac{27 q^{6} r^{13} p^{6}}{8 p^{3} {(q^{4})}^{2}}$

Step 8

Reduce the expression by cancelling the common factors.

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Step 8.1

Multiply the exponents in ${(q^{4})}^{2}$ .

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Step 8.1.1

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

$\frac{27 q^{6} r^{13} p^{6}}{8 p^{3} q^{4 \cdot 2}}$

Step 8.1.2

Multiply $4$ by $2$ .

$\frac{27 q^{6} r^{13} p^{6}}{8 p^{3} q^{8}}$

Step 8.2

Cancel the common factor of $q^{6}$ and $q^{8}$ .

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Step 8.2.1

Factor $q^{6}$ out of $27 q^{6} r^{13} p^{6}$ .

$\frac{q^{6} (27 r^{13} p^{6})}{8 p^{3} q^{8}}$

Step 8.2.2

Cancel the common factors.

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Step 8.2.2.1

Factor $q^{6}$ out of $8 p^{3} q^{8}$ .

$\frac{q^{6} (27 r^{13} p^{6})}{q^{6} (8 p^{3} q^{2})}$

Step 8.2.2.2

Cancel the common factor.

$\frac{q^{6} (27 r^{13} p^{6})}{q^{6} (8 p^{3} q^{2})}$

Step 8.2.2.3

Rewrite the expression.

$\frac{27 r^{13} p^{6}}{8 p^{3} q^{2}}$

Step 8.3

Cancel the common factor of $p^{6}$ and $p^{3}$ .

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Step 8.3.1

Factor $p^{3}$ out of $27 r^{13} p^{6}$ .

$\frac{p^{3} (27 r^{13} p^{3})}{8 p^{3} q^{2}}$

Step 8.3.2

Cancel the common factors.

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Step 8.3.2.1

Factor $p^{3}$ out of $8 p^{3} q^{2}$ .

$\frac{p^{3} (27 r^{13} p^{3})}{p^{3} (8 q^{2})}$

Step 8.3.2.2

Cancel the common factor.

$\frac{p^{3} (27 r^{13} p^{3})}{p^{3} (8 q^{2})}$

Step 8.3.2.3

Rewrite the expression.

$\frac{27 r^{13} p^{3}}{8 q^{2}}$