Algebra Examples

${(\frac{6 a^{3} b^{4}}{z^{2} y^{4}})}^{- 3} \cdot {(\frac{2 z y}{3 a b})}^{- 2}$

Step 1

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

${(\frac{z^{2} y^{4}}{6 a^{3} b^{4}})}^{3} \cdot {(\frac{2 z y}{3 a b})}^{- 2}$

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 $\frac{z^{2} y^{4}}{6 a^{3} b^{4}}$ .

$\frac{{(z^{2} y^{4})}^{3}}{{(6 a^{3} b^{4})}^{3}} \cdot {(\frac{2 z y}{3 a b})}^{- 2}$

Step 2.2

Apply the product rule to $z^{2} y^{4}$ .

$\frac{{(z^{2})}^{3} {(y^{4})}^{3}}{{(6 a^{3} b^{4})}^{3}} \cdot {(\frac{2 z y}{3 a b})}^{- 2}$

Step 2.3

Apply the product rule to $6 a^{3} b^{4}$ .

$\frac{{(z^{2})}^{3} {(y^{4})}^{3}}{{(6 a^{3})}^{3} {(b^{4})}^{3}} \cdot {(\frac{2 z y}{3 a b})}^{- 2}$

Step 2.4

Apply the product rule to $6 a^{3}$ .

$\frac{{(z^{2})}^{3} {(y^{4})}^{3}}{6^{3} {(a^{3})}^{3} {(b^{4})}^{3}} \cdot {(\frac{2 z y}{3 a b})}^{- 2}$

Step 3

Simplify the numerator.

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Step 3.1

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

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Step 3.1.1

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

$\frac{z^{2 \cdot 3} {(y^{4})}^{3}}{6^{3} {(a^{3})}^{3} {(b^{4})}^{3}} \cdot {(\frac{2 z y}{3 a b})}^{- 2}$

Step 3.1.2

Multiply $2$ by $3$ .

$\frac{z^{6} {(y^{4})}^{3}}{6^{3} {(a^{3})}^{3} {(b^{4})}^{3}} \cdot {(\frac{2 z y}{3 a b})}^{- 2}$

Step 3.2

Multiply the exponents in ${(y^{4})}^{3}$ .

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Step 3.2.1

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

$\frac{z^{6} y^{4 \cdot 3}}{6^{3} {(a^{3})}^{3} {(b^{4})}^{3}} \cdot {(\frac{2 z y}{3 a b})}^{- 2}$

Step 3.2.2

Multiply $4$ by $3$ .

$\frac{z^{6} y^{12}}{6^{3} {(a^{3})}^{3} {(b^{4})}^{3}} \cdot {(\frac{2 z y}{3 a b})}^{- 2}$

Step 4

Simplify the denominator.

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Step 4.1

Raise $6$ to the power of $3$ .

$\frac{z^{6} y^{12}}{216 {(a^{3})}^{3} {(b^{4})}^{3}} \cdot {(\frac{2 z y}{3 a b})}^{- 2}$

Step 4.2

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

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Step 4.2.1

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

$\frac{z^{6} y^{12}}{216 a^{3 \cdot 3} {(b^{4})}^{3}} \cdot {(\frac{2 z y}{3 a b})}^{- 2}$

Step 4.2.2

Multiply $3$ by $3$ .

$\frac{z^{6} y^{12}}{216 a^{9} {(b^{4})}^{3}} \cdot {(\frac{2 z y}{3 a b})}^{- 2}$

Step 4.3

Multiply the exponents in ${(b^{4})}^{3}$ .

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Step 4.3.1

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

$\frac{z^{6} y^{12}}{216 a^{9} b^{4 \cdot 3}} \cdot {(\frac{2 z y}{3 a b})}^{- 2}$

Step 4.3.2

Multiply $4$ by $3$ .

$\frac{z^{6} y^{12}}{216 a^{9} b^{12}} \cdot {(\frac{2 z y}{3 a b})}^{- 2}$

Step 5

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

$\frac{z^{6} y^{12}}{216 a^{9} b^{12}} \cdot {(\frac{3 a b}{2 z y})}^{2}$

Step 6

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

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Step 6.1

Apply the product rule to $\frac{3 a b}{2 z y}$ .

$\frac{z^{6} y^{12}}{216 a^{9} b^{12}} \cdot \frac{{(3 a b)}^{2}}{{(2 z y)}^{2}}$

Step 6.2

Apply the product rule to $3 a b$ .

$\frac{z^{6} y^{12}}{216 a^{9} b^{12}} \cdot \frac{{(3 a)}^{2} b^{2}}{{(2 z y)}^{2}}$

Step 6.3

Apply the product rule to $3 a$ .

$\frac{z^{6} y^{12}}{216 a^{9} b^{12}} \cdot \frac{3^{2} a^{2} b^{2}}{{(2 z y)}^{2}}$

Step 6.4

Apply the product rule to $2 z y$ .

$\frac{z^{6} y^{12}}{216 a^{9} b^{12}} \cdot \frac{3^{2} a^{2} b^{2}}{{(2 z)}^{2} y^{2}}$

Step 6.5

Apply the product rule to $2 z$ .

$\frac{z^{6} y^{12}}{216 a^{9} b^{12}} \cdot \frac{3^{2} a^{2} b^{2}}{2^{2} z^{2} y^{2}}$

Step 7

Simplify terms.

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Step 7.1

Combine.

$\frac{z^{6} y^{12} (3^{2} a^{2} b^{2})}{216 a^{9} b^{12} (2^{2} z^{2} y^{2})}$

Step 7.2

Cancel the common factor of $z^{6}$ and $z^{2}$ .

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Step 7.2.1

Factor $z^{2}$ out of $z^{6} y^{12} (3^{2} a^{2} b^{2})$ .

$\frac{z^{2} (z^{4} y^{12} (3^{2} a^{2} b^{2}))}{216 a^{9} b^{12} (2^{2} z^{2} y^{2})}$

Step 7.2.2

Cancel the common factors.

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Step 7.2.2.1

Factor $z^{2}$ out of $216 a^{9} b^{12} (2^{2} z^{2} y^{2})$ .

$\frac{z^{2} (z^{4} y^{12} (3^{2} a^{2} b^{2}))}{z^{2} (216 a^{9} b^{12} (2^{2} y^{2}))}$

Step 7.2.2.2

Cancel the common factor.

$\frac{z^{2} (z^{4} y^{12} (3^{2} a^{2} b^{2}))}{z^{2} (216 a^{9} b^{12} (2^{2} y^{2}))}$

Step 7.2.2.3

Rewrite the expression.

$\frac{z^{4} y^{12} (3^{2} a^{2} b^{2})}{216 a^{9} b^{12} (2^{2} y^{2})}$

Step 7.3

Cancel the common factor of $y^{12}$ and $y^{2}$ .

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Step 7.3.1

Factor $y^{2}$ out of $z^{4} y^{12} (3^{2} a^{2} b^{2})$ .

$\frac{y^{2} (z^{4} y^{10} (3^{2} a^{2} b^{2}))}{216 a^{9} b^{12} (2^{2} y^{2})}$

Step 7.3.2

Cancel the common factors.

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Step 7.3.2.1

Factor $y^{2}$ out of $216 a^{9} b^{12} (2^{2} y^{2})$ .

$\frac{y^{2} (z^{4} y^{10} (3^{2} a^{2} b^{2}))}{y^{2} (216 a^{9} b^{12} \cdot (2^{2}))}$

Step 7.3.2.2

Cancel the common factor.

$\frac{y^{2} (z^{4} y^{10} (3^{2} a^{2} b^{2}))}{y^{2} (216 a^{9} b^{12} \cdot (2^{2}))}$

Step 7.3.2.3

Rewrite the expression.

$\frac{z^{4} y^{10} (3^{2} a^{2} b^{2})}{216 a^{9} b^{12} \cdot (2^{2})}$

Step 7.4

Cancel the common factor of $a^{2}$ and $a^{9}$ .

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Step 7.4.1

Factor $a^{2}$ out of $z^{4} y^{10} (3^{2} a^{2} b^{2})$ .

$\frac{a^{2} (z^{4} y^{10} (3^{2} b^{2}))}{216 a^{9} b^{12} \cdot (2^{2})}$

Step 7.4.2

Cancel the common factors.

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Step 7.4.2.1

Factor $a^{2}$ out of $216 a^{9} b^{12} \cdot (2^{2})$ .

$\frac{a^{2} (z^{4} y^{10} (3^{2} b^{2}))}{a^{2} (216 a^{7} b^{12} \cdot 2^{2})}$

Step 7.4.2.2

Cancel the common factor.

$\frac{a^{2} (z^{4} y^{10} (3^{2} b^{2}))}{a^{2} (216 a^{7} b^{12} \cdot 2^{2})}$

Step 7.4.2.3

Rewrite the expression.

$\frac{z^{4} y^{10} (3^{2} b^{2})}{216 a^{7} b^{12} \cdot 2^{2}}$

Step 7.5

Cancel the common factor of $b^{2}$ and $b^{12}$ .

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Step 7.5.1

Factor $b^{2}$ out of $z^{4} y^{10} (3^{2} b^{2})$ .

$\frac{b^{2} (z^{4} y^{10} \cdot (3^{2}))}{216 a^{7} b^{12} \cdot 2^{2}}$

Step 7.5.2

Cancel the common factors.

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Step 7.5.2.1

Factor $b^{2}$ out of $216 a^{7} b^{12} \cdot 2^{2}$ .

$\frac{b^{2} (z^{4} y^{10} \cdot (3^{2}))}{b^{2} (216 a^{7} b^{10} \cdot 2^{2})}$

Step 7.5.2.2

Cancel the common factor.

$\frac{b^{2} (z^{4} y^{10} \cdot (3^{2}))}{b^{2} (216 a^{7} b^{10} \cdot 2^{2})}$

Step 7.5.2.3

Rewrite the expression.

$\frac{z^{4} y^{10} \cdot (3^{2})}{216 a^{7} b^{10} \cdot 2^{2}}$

Step 7.6

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

$\frac{z^{4} y^{10} \cdot 9}{216 a^{7} b^{10} \cdot 2^{2}}$

Step 8

Simplify the denominator.

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Step 8.1

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

$\frac{z^{4} y^{10} \cdot 9}{216 a^{7} b^{10} \cdot 4}$

Step 8.2

Multiply $4$ by $216$ .

$\frac{z^{4} y^{10} \cdot 9}{864 a^{7} b^{10}}$

Step 9

Cancel the common factor of $9$ and $864$ .

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Step 9.1

Factor $9$ out of $z^{4} y^{10} \cdot 9$ .

$\frac{9 \cdot (z^{4} y^{10})}{864 a^{7} b^{10}}$

Step 9.2

Cancel the common factors.

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Step 9.2.1

Factor $9$ out of $864 a^{7} b^{10}$ .

$\frac{9 \cdot (z^{4} y^{10})}{9 (96 a^{7} b^{10})}$

Step 9.2.2

Cancel the common factor.

$\frac{9 \cdot (z^{4} y^{10})}{9 (96 a^{7} b^{10})}$

Step 9.2.3

Rewrite the expression.

$\frac{z^{4} y^{10}}{96 a^{7} b^{10}}$