Algebra Examples

${(\frac{{(2 x^{4} y^{- 3})}^{- 2} \cdot {(2 x^{- 1} y^{- 2})}^{4}}{2 x^{- 7} y^{- 3}})}^{3}$

Step 1

Simplify.

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Step 1.1

Move ${(2 x^{4} y^{- 3})}^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${(\frac{{(2 x^{- 1} y^{- 2})}^{4}}{2 x^{- 7} y^{- 3} {(2 x^{4} y^{- 3})}^{2}})}^{3}$

Step 1.2

Move $x^{- 7}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

${(\frac{{(2 x^{- 1} y^{- 2})}^{4} x^{7}}{2 y^{- 3} {(2 x^{4} y^{- 3})}^{2}})}^{3}$

Step 1.3

Move $y^{- 3}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

${(\frac{{(2 x^{- 1} y^{- 2})}^{4} x^{7} y^{3}}{2 {(2 x^{4} y^{- 3})}^{2}})}^{3}$

Step 2

Simplify the numerator.

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Step 2.1

Apply the product rule to $2 x^{- 1} y^{- 2}$ .

${(\frac{{(2 x^{- 1})}^{4} {(y^{- 2})}^{4} x^{7} y^{3}}{2 {(2 x^{4} y^{- 3})}^{2}})}^{3}$

Step 2.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${(\frac{{(2 \frac{1}{x})}^{4} {(y^{- 2})}^{4} x^{7} y^{3}}{2 {(2 x^{4} y^{- 3})}^{2}})}^{3}$

Step 2.3

Combine $2$ and $\frac{1}{x}$ .

${(\frac{{(\frac{2}{x})}^{4} {(y^{- 2})}^{4} x^{7} y^{3}}{2 {(2 x^{4} y^{- 3})}^{2}})}^{3}$

Step 2.4

Apply the product rule to $\frac{2}{x}$ .

${(\frac{\frac{2^{4}}{x^{4}} {(y^{- 2})}^{4} x^{7} y^{3}}{2 {(2 x^{4} y^{- 3})}^{2}})}^{3}$

Step 2.5

Raise $2$ to the power of $4$ .

${(\frac{\frac{16}{x^{4}} {(y^{- 2})}^{4} x^{7} y^{3}}{2 {(2 x^{4} y^{- 3})}^{2}})}^{3}$

Step 2.6

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

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Step 2.6.1

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

${(\frac{\frac{16}{x^{4}} y^{- 2 \cdot 4} x^{7} y^{3}}{2 {(2 x^{4} y^{- 3})}^{2}})}^{3}$

Step 2.6.2

Multiply $- 2$ by $4$ .

${(\frac{\frac{16}{x^{4}} y^{- 8} x^{7} y^{3}}{2 {(2 x^{4} y^{- 3})}^{2}})}^{3}$

Step 2.7

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${(\frac{\frac{16}{x^{4}} \cdot \frac{1}{y^{8}} x^{7} y^{3}}{2 {(2 x^{4} y^{- 3})}^{2}})}^{3}$

Step 2.8

Combine exponents.

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Step 2.8.1

Multiply $\frac{16}{x^{4}}$ by $\frac{1}{y^{8}}$ .

${(\frac{\frac{16}{x^{4} y^{8}} x^{7} y^{3}}{2 {(2 x^{4} y^{- 3})}^{2}})}^{3}$

Step 2.8.2

Combine $\frac{16}{x^{4} y^{8}}$ and $x^{7}$ .

${(\frac{\frac{16 x^{7}}{x^{4} y^{8}} y^{3}}{2 {(2 x^{4} y^{- 3})}^{2}})}^{3}$

Step 2.8.3

Combine $\frac{16 x^{7}}{x^{4} y^{8}}$ and $y^{3}$ .

${(\frac{\frac{16 x^{7} y^{3}}{x^{4} y^{8}}}{2 {(2 x^{4} y^{- 3})}^{2}})}^{3}$

Step 2.9

Reduce the expression $\frac{16 x^{7} y^{3}}{x^{4} y^{8}}$ by cancelling the common factors.

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Step 2.9.1

Factor $x^{4}$ out of $16 x^{7} y^{3}$ .

${(\frac{\frac{x^{4} (16 x^{3} y^{3})}{x^{4} y^{8}}}{2 {(2 x^{4} y^{- 3})}^{2}})}^{3}$

Step 2.9.2

Factor $x^{4}$ out of $x^{4} y^{8}$ .

${(\frac{\frac{x^{4} (16 x^{3} y^{3})}{x^{4} (y^{8})}}{2 {(2 x^{4} y^{- 3})}^{2}})}^{3}$

Step 2.9.3

Cancel the common factor.

${(\frac{\frac{x^{4} (16 x^{3} y^{3})}{x^{4} y^{8}}}{2 {(2 x^{4} y^{- 3})}^{2}})}^{3}$

Step 2.9.4

Rewrite the expression.

${(\frac{\frac{16 x^{3} y^{3}}{y^{8}}}{2 {(2 x^{4} y^{- 3})}^{2}})}^{3}$

Step 2.10

Cancel the common factor of $y^{3}$ and $y^{8}$ .

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Step 2.10.1

Factor $y^{3}$ out of $16 x^{3} y^{3}$ .

${(\frac{\frac{y^{3} (16 x^{3})}{y^{8}}}{2 {(2 x^{4} y^{- 3})}^{2}})}^{3}$

Step 2.10.2

Cancel the common factors.

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Step 2.10.2.1

Factor $y^{3}$ out of $y^{8}$ .

${(\frac{\frac{y^{3} (16 x^{3})}{y^{3} y^{5}}}{2 {(2 x^{4} y^{- 3})}^{2}})}^{3}$

Step 2.10.2.2

Cancel the common factor.

${(\frac{\frac{y^{3} (16 x^{3})}{y^{3} y^{5}}}{2 {(2 x^{4} y^{- 3})}^{2}})}^{3}$

Step 2.10.2.3

Rewrite the expression.

${(\frac{\frac{16 x^{3}}{y^{5}}}{2 {(2 x^{4} y^{- 3})}^{2}})}^{3}$

Step 3

Simplify the denominator.

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Step 3.1

Apply the product rule to $2 x^{4} y^{- 3}$ .

${(\frac{\frac{16 x^{3}}{y^{5}}}{2 {(2 x^{4})}^{2} {(y^{- 3})}^{2}})}^{3}$

Step 3.2

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

${(\frac{\frac{16 x^{3}}{y^{5}}}{2 (2^{2} {(x^{4})}^{2}) {(y^{- 3})}^{2}})}^{3}$

Step 3.3

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

${(\frac{\frac{16 x^{3}}{y^{5}}}{2 (4 {(x^{4})}^{2}) {(y^{- 3})}^{2}})}^{3}$

Step 3.4

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

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Step 3.4.1

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

${(\frac{\frac{16 x^{3}}{y^{5}}}{2 (4 x^{4 \cdot 2}) {(y^{- 3})}^{2}})}^{3}$

Step 3.4.2

Multiply $4$ by $2$ .

${(\frac{\frac{16 x^{3}}{y^{5}}}{2 (4 x^{8}) {(y^{- 3})}^{2}})}^{3}$

Step 3.5

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

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Step 3.5.1

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

${(\frac{\frac{16 x^{3}}{y^{5}}}{2 (4 x^{8}) y^{- 3 \cdot 2}})}^{3}$

Step 3.5.2

Multiply $- 3$ by $2$ .

${(\frac{\frac{16 x^{3}}{y^{5}}}{2 (4 x^{8}) y^{- 6}})}^{3}$

Step 3.6

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${(\frac{\frac{16 x^{3}}{y^{5}}}{2 \cdot 4 x^{8} \frac{1}{y^{6}}})}^{3}$

Step 3.7

Combine exponents.

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Step 3.7.1

Multiply $2$ by $4$ .

${(\frac{\frac{16 x^{3}}{y^{5}}}{8 x^{8} \frac{1}{y^{6}}})}^{3}$

Step 3.7.2

Combine $8$ and $\frac{1}{y^{6}}$ .

${(\frac{\frac{16 x^{3}}{y^{5}}}{x^{8} \frac{8}{y^{6}}})}^{3}$

Step 3.7.3

Combine $x^{8}$ and $\frac{8}{y^{6}}$ .

${(\frac{\frac{16 x^{3}}{y^{5}}}{\frac{x^{8} \cdot 8}{y^{6}}})}^{3}$

Step 3.8

Move $8$ to the left of $x^{8}$ .

${(\frac{\frac{16 x^{3}}{y^{5}}}{\frac{8 x^{8}}{y^{6}}})}^{3}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

${(\frac{16 x^{3}}{y^{5}} \cdot \frac{y^{6}}{8 x^{8}})}^{3}$

Step 5

Combine.

${(\frac{16 x^{3} y^{6}}{y^{5} (8 x^{8})})}^{3}$

Step 6

Cancel the common factor of $16$ and $8$ .

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Step 6.1

Factor $8$ out of $16 x^{3} y^{6}$ .

${(\frac{8 (2 x^{3} y^{6})}{y^{5} (8 x^{8})})}^{3}$

Step 6.2

Cancel the common factors.

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Step 6.2.1

Factor $8$ out of $y^{5} (8 x^{8})$ .

${(\frac{8 (2 x^{3} y^{6})}{8 (y^{5} (x^{8}))})}^{3}$

Step 6.2.2

Cancel the common factor.

${(\frac{8 (2 x^{3} y^{6})}{8 (y^{5} (x^{8}))})}^{3}$

Step 6.2.3

Rewrite the expression.

${(\frac{2 x^{3} y^{6}}{y^{5} (x^{8})})}^{3}$

Step 7

Cancel the common factor of $x^{3}$ and $x^{8}$ .

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Step 7.1

Factor $x^{3}$ out of $2 x^{3} y^{6}$ .

${(\frac{x^{3} (2 y^{6})}{y^{5} (x^{8})})}^{3}$

Step 7.2

Cancel the common factors.

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Step 7.2.1

Factor $x^{3}$ out of $y^{5} (x^{8})$ .

${(\frac{x^{3} (2 y^{6})}{x^{3} (y^{5} (x^{5}))})}^{3}$

Step 7.2.2

Cancel the common factor.

${(\frac{x^{3} (2 y^{6})}{x^{3} (y^{5} (x^{5}))})}^{3}$

Step 7.2.3

Rewrite the expression.

${(\frac{2 y^{6}}{y^{5} (x^{5})})}^{3}$

Step 8

Cancel the common factor of $y^{6}$ and $y^{5}$ .

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Step 8.1

Factor $y^{5}$ out of $2 y^{6}$ .

${(\frac{y^{5} (2 y)}{y^{5} (x^{5})})}^{3}$

Step 8.2

Cancel the common factors.

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Step 8.2.1

Cancel the common factor.

${(\frac{y^{5} (2 y)}{y^{5} x^{5}})}^{3}$

Step 8.2.2

Rewrite the expression.

${(\frac{2 y}{x^{5}})}^{3}$

Step 9

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

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Step 9.1

Apply the product rule to $\frac{2 y}{x^{5}}$ .

$\frac{{(2 y)}^{3}}{{(x^{5})}^{3}}$

Step 9.2

Apply the product rule to $2 y$ .

$\frac{2^{3} y^{3}}{{(x^{5})}^{3}}$

Step 10

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

$\frac{8 y^{3}}{{(x^{5})}^{3}}$

Step 11

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

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Step 11.1

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

$\frac{8 y^{3}}{x^{5 \cdot 3}}$

Step 11.2

Multiply $5$ by $3$ .

$\frac{8 y^{3}}{x^{15}}$