Algebra Examples

${(27 x^{12} y^{- 6} z^{3})}^{\frac{2}{3}} \cdot {(64 x^{4} y^{6} z^{- 2})}^{- \frac{1}{2}}$

Step 1

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

${(27 x^{12} \frac{1}{y^{6}} z^{3})}^{\frac{2}{3}} \cdot {(64 x^{4} y^{6} z^{- 2})}^{- \frac{1}{2}}$

Step 2

Multiply $27 x^{12} \frac{1}{y^{6}}$ .

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Step 2.1

Combine $27$ and $\frac{1}{y^{6}}$ .

${(x^{12} \frac{27}{y^{6}} z^{3})}^{\frac{2}{3}} \cdot {(64 x^{4} y^{6} z^{- 2})}^{- \frac{1}{2}}$

Step 2.2

Combine $x^{12}$ and $\frac{27}{y^{6}}$ .

${(\frac{x^{12} \cdot 27}{y^{6}} z^{3})}^{\frac{2}{3}} \cdot {(64 x^{4} y^{6} z^{- 2})}^{- \frac{1}{2}}$

Step 3

Move $27$ to the left of $x^{12}$ .

${(\frac{27 \cdot x^{12}}{y^{6}} z^{3})}^{\frac{2}{3}} \cdot {(64 x^{4} y^{6} z^{- 2})}^{- \frac{1}{2}}$

Step 4

Combine $\frac{27 x^{12}}{y^{6}}$ and $z^{3}$ .

${(\frac{27 x^{12} z^{3}}{y^{6}})}^{\frac{2}{3}} \cdot {(64 x^{4} y^{6} z^{- 2})}^{- \frac{1}{2}}$

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 $\frac{27 x^{12} z^{3}}{y^{6}}$ .

$\frac{{(27 x^{12} z^{3})}^{\frac{2}{3}}}{{(y^{6})}^{\frac{2}{3}}} \cdot {(64 x^{4} y^{6} z^{- 2})}^{- \frac{1}{2}}$

Step 5.2

Apply the product rule to $27 x^{12} z^{3}$ .

$\frac{{(27 x^{12})}^{\frac{2}{3}} {(z^{3})}^{\frac{2}{3}}}{{(y^{6})}^{\frac{2}{3}}} \cdot {(64 x^{4} y^{6} z^{- 2})}^{- \frac{1}{2}}$

Step 5.3

Apply the product rule to $27 x^{12}$ .

$\frac{27^{\frac{2}{3}} {(x^{12})}^{\frac{2}{3}} {(z^{3})}^{\frac{2}{3}}}{{(y^{6})}^{\frac{2}{3}}} \cdot {(64 x^{4} y^{6} z^{- 2})}^{- \frac{1}{2}}$

Step 6

Simplify the numerator.

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Step 6.1

Rewrite $27$ as $3^{3}$ .

$\frac{{(3^{3})}^{\frac{2}{3}} {(x^{12})}^{\frac{2}{3}} {(z^{3})}^{\frac{2}{3}}}{{(y^{6})}^{\frac{2}{3}}} \cdot {(64 x^{4} y^{6} z^{- 2})}^{- \frac{1}{2}}$

Step 6.2

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

$\frac{3^{3 (\frac{2}{3})} {(x^{12})}^{\frac{2}{3}} {(z^{3})}^{\frac{2}{3}}}{{(y^{6})}^{\frac{2}{3}}} \cdot {(64 x^{4} y^{6} z^{- 2})}^{- \frac{1}{2}}$

Step 6.3

Cancel the common factor of $3$ .

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Step 6.3.1

Cancel the common factor.

$\frac{3^{3 (\frac{2}{3})} {(x^{12})}^{\frac{2}{3}} {(z^{3})}^{\frac{2}{3}}}{{(y^{6})}^{\frac{2}{3}}} \cdot {(64 x^{4} y^{6} z^{- 2})}^{- \frac{1}{2}}$

Step 6.3.2

Rewrite the expression.

$\frac{3^{2} {(x^{12})}^{\frac{2}{3}} {(z^{3})}^{\frac{2}{3}}}{{(y^{6})}^{\frac{2}{3}}} \cdot {(64 x^{4} y^{6} z^{- 2})}^{- \frac{1}{2}}$

Step 6.4

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

$\frac{9 {(x^{12})}^{\frac{2}{3}} {(z^{3})}^{\frac{2}{3}}}{{(y^{6})}^{\frac{2}{3}}} \cdot {(64 x^{4} y^{6} z^{- 2})}^{- \frac{1}{2}}$

Step 6.5

Multiply the exponents in ${(x^{12})}^{\frac{2}{3}}$ .

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Step 6.5.1

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

$\frac{9 x^{12 (\frac{2}{3})} {(z^{3})}^{\frac{2}{3}}}{{(y^{6})}^{\frac{2}{3}}} \cdot {(64 x^{4} y^{6} z^{- 2})}^{- \frac{1}{2}}$

Step 6.5.2

Cancel the common factor of $3$ .

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Step 6.5.2.1

Factor $3$ out of $12$ .

$\frac{9 x^{3 (4) \frac{2}{3}} {(z^{3})}^{\frac{2}{3}}}{{(y^{6})}^{\frac{2}{3}}} \cdot {(64 x^{4} y^{6} z^{- 2})}^{- \frac{1}{2}}$

Step 6.5.2.2

Cancel the common factor.

$\frac{9 x^{3 \cdot 4 \frac{2}{3}} {(z^{3})}^{\frac{2}{3}}}{{(y^{6})}^{\frac{2}{3}}} \cdot {(64 x^{4} y^{6} z^{- 2})}^{- \frac{1}{2}}$

Step 6.5.2.3

Rewrite the expression.

$\frac{9 x^{4 \cdot 2} {(z^{3})}^{\frac{2}{3}}}{{(y^{6})}^{\frac{2}{3}}} \cdot {(64 x^{4} y^{6} z^{- 2})}^{- \frac{1}{2}}$

Step 6.5.3

Multiply $4$ by $2$ .

$\frac{9 x^{8} {(z^{3})}^{\frac{2}{3}}}{{(y^{6})}^{\frac{2}{3}}} \cdot {(64 x^{4} y^{6} z^{- 2})}^{- \frac{1}{2}}$

Step 6.6

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

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Step 6.6.1

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

$\frac{9 x^{8} z^{3 (\frac{2}{3})}}{{(y^{6})}^{\frac{2}{3}}} \cdot {(64 x^{4} y^{6} z^{- 2})}^{- \frac{1}{2}}$

Step 6.6.2

Cancel the common factor of $3$ .

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Step 6.6.2.1

Cancel the common factor.

$\frac{9 x^{8} z^{3 (\frac{2}{3})}}{{(y^{6})}^{\frac{2}{3}}} \cdot {(64 x^{4} y^{6} z^{- 2})}^{- \frac{1}{2}}$

Step 6.6.2.2

Rewrite the expression.

$\frac{9 x^{8} z^{2}}{{(y^{6})}^{\frac{2}{3}}} \cdot {(64 x^{4} y^{6} z^{- 2})}^{- \frac{1}{2}}$

Step 7

Multiply the exponents in ${(y^{6})}^{\frac{2}{3}}$ .

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Step 7.1

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

$\frac{9 x^{8} z^{2}}{y^{6 (\frac{2}{3})}} \cdot {(64 x^{4} y^{6} z^{- 2})}^{- \frac{1}{2}}$

Step 7.2

Cancel the common factor of $3$ .

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Step 7.2.1

Factor $3$ out of $6$ .

$\frac{9 x^{8} z^{2}}{y^{3 (2) \frac{2}{3}}} \cdot {(64 x^{4} y^{6} z^{- 2})}^{- \frac{1}{2}}$

Step 7.2.2

Cancel the common factor.

$\frac{9 x^{8} z^{2}}{y^{3 \cdot 2 \frac{2}{3}}} \cdot {(64 x^{4} y^{6} z^{- 2})}^{- \frac{1}{2}}$

Step 7.2.3

Rewrite the expression.

$\frac{9 x^{8} z^{2}}{y^{2 \cdot 2}} \cdot {(64 x^{4} y^{6} z^{- 2})}^{- \frac{1}{2}}$

Step 7.3

Multiply $2$ by $2$ .

$\frac{9 x^{8} z^{2}}{y^{4}} \cdot {(64 x^{4} y^{6} z^{- 2})}^{- \frac{1}{2}}$

Step 8

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

$\frac{9 x^{8} z^{2}}{y^{4}} \cdot {(64 x^{4} y^{6} \frac{1}{z^{2}})}^{- \frac{1}{2}}$

Step 9

Multiply $64 x^{4} y^{6} \frac{1}{z^{2}}$ .

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Step 9.1

Combine $64$ and $\frac{1}{z^{2}}$ .

$\frac{9 x^{8} z^{2}}{y^{4}} \cdot {(x^{4} y^{6} \frac{64}{z^{2}})}^{- \frac{1}{2}}$

Step 9.2

Combine $x^{4}$ and $\frac{64}{z^{2}}$ .

$\frac{9 x^{8} z^{2}}{y^{4}} \cdot {(y^{6} \frac{x^{4} \cdot 64}{z^{2}})}^{- \frac{1}{2}}$

Step 9.3

Combine $y^{6}$ and $\frac{x^{4} \cdot 64}{z^{2}}$ .

$\frac{9 x^{8} z^{2}}{y^{4}} \cdot {(\frac{y^{6} (x^{4} \cdot 64)}{z^{2}})}^{- \frac{1}{2}}$

Step 10

Move $64$ to the left of $y^{6} x^{4}$ .

$\frac{9 x^{8} z^{2}}{y^{4}} \cdot {(\frac{64 \cdot (y^{6} x^{4})}{z^{2}})}^{- \frac{1}{2}}$

Step 11

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

$\frac{9 x^{8} z^{2}}{y^{4}} \cdot {(\frac{z^{2}}{64 y^{6} x^{4}})}^{\frac{1}{2}}$

Step 12

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

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Step 12.1

Apply the product rule to $\frac{z^{2}}{64 y^{6} x^{4}}$ .

$\frac{9 x^{8} z^{2}}{y^{4}} \cdot \frac{{(z^{2})}^{\frac{1}{2}}}{{(64 y^{6} x^{4})}^{\frac{1}{2}}}$

Step 12.2

Apply the product rule to $64 y^{6} x^{4}$ .

$\frac{9 x^{8} z^{2}}{y^{4}} \cdot \frac{{(z^{2})}^{\frac{1}{2}}}{{(64 y^{6})}^{\frac{1}{2}} {(x^{4})}^{\frac{1}{2}}}$

Step 12.3

Apply the product rule to $64 y^{6}$ .

$\frac{9 x^{8} z^{2}}{y^{4}} \cdot \frac{{(z^{2})}^{\frac{1}{2}}}{64^{\frac{1}{2}} {(y^{6})}^{\frac{1}{2}} {(x^{4})}^{\frac{1}{2}}}$

Step 13

Combine.

$\frac{9 x^{8} z^{2} {(z^{2})}^{\frac{1}{2}}}{y^{4} (64^{\frac{1}{2}} {(y^{6})}^{\frac{1}{2}} {(x^{4})}^{\frac{1}{2}})}$

Step 14

Simplify the numerator.

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Step 14.1

Multiply the exponents in ${(z^{2})}^{\frac{1}{2}}$ .

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Step 14.1.1

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

$\frac{9 x^{8} z^{2} z^{2 (\frac{1}{2})}}{y^{4} (64^{\frac{1}{2}} {(y^{6})}^{\frac{1}{2}} {(x^{4})}^{\frac{1}{2}})}$

Step 14.1.2

Cancel the common factor of $2$ .

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Step 14.1.2.1

Cancel the common factor.

$\frac{9 x^{8} z^{2} z^{2 (\frac{1}{2})}}{y^{4} (64^{\frac{1}{2}} {(y^{6})}^{\frac{1}{2}} {(x^{4})}^{\frac{1}{2}})}$

Step 14.1.2.2

Rewrite the expression.

$\frac{9 x^{8} z^{2} z^{1}}{y^{4} (64^{\frac{1}{2}} {(y^{6})}^{\frac{1}{2}} {(x^{4})}^{\frac{1}{2}})}$

Step 14.2

Simplify.

$\frac{9 x^{8} z^{2} z}{y^{4} (64^{\frac{1}{2}} {(y^{6})}^{\frac{1}{2}} {(x^{4})}^{\frac{1}{2}})}$

Step 14.3

Combine exponents.

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Step 14.3.1

Raise $z$ to the power of $1$ .

$\frac{9 x^{8} (z^{1} z^{2})}{y^{4} (64^{\frac{1}{2}} {(y^{6})}^{\frac{1}{2}} {(x^{4})}^{\frac{1}{2}})}$

Step 14.3.2

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

$\frac{9 x^{8} z^{1 + 2}}{y^{4} (64^{\frac{1}{2}} {(y^{6})}^{\frac{1}{2}} {(x^{4})}^{\frac{1}{2}})}$

Step 14.3.3

Add $1$ and $2$ .

$\frac{9 x^{8} z^{3}}{y^{4} (64^{\frac{1}{2}} {(y^{6})}^{\frac{1}{2}} {(x^{4})}^{\frac{1}{2}})}$

Step 15

Simplify the denominator.

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Step 15.1

Remove unnecessary parentheses.

$\frac{9 x^{8} z^{3}}{y^{4} \cdot 64^{\frac{1}{2}} {(y^{6})}^{\frac{1}{2}} {(x^{4})}^{\frac{1}{2}}}$

Step 15.2

Rewrite $64$ as $8^{2}$ .

$\frac{9 x^{8} z^{3}}{y^{4} \cdot {(8^{2})}^{\frac{1}{2}} {(y^{6})}^{\frac{1}{2}} {(x^{4})}^{\frac{1}{2}}}$

Step 15.3

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

$\frac{9 x^{8} z^{3}}{y^{4} \cdot 8^{2 (\frac{1}{2})} {(y^{6})}^{\frac{1}{2}} {(x^{4})}^{\frac{1}{2}}}$

Step 15.4

Cancel the common factor of $2$ .

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Step 15.4.1

Cancel the common factor.

$\frac{9 x^{8} z^{3}}{y^{4} \cdot 8^{2 (\frac{1}{2})} {(y^{6})}^{\frac{1}{2}} {(x^{4})}^{\frac{1}{2}}}$

Step 15.4.2

Rewrite the expression.

$\frac{9 x^{8} z^{3}}{y^{4} \cdot 8^{1} {(y^{6})}^{\frac{1}{2}} {(x^{4})}^{\frac{1}{2}}}$

Step 15.5

Evaluate the exponent.

$\frac{9 x^{8} z^{3}}{y^{4} \cdot 8 {(y^{6})}^{\frac{1}{2}} {(x^{4})}^{\frac{1}{2}}}$

Step 15.6

Multiply the exponents in ${(y^{6})}^{\frac{1}{2}}$ .

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Step 15.6.1

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

$\frac{9 x^{8} z^{3}}{y^{4} \cdot 8 y^{6 (\frac{1}{2})} {(x^{4})}^{\frac{1}{2}}}$

Step 15.6.2

Cancel the common factor of $2$ .

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Step 15.6.2.1

Factor $2$ out of $6$ .

$\frac{9 x^{8} z^{3}}{y^{4} \cdot 8 y^{2 (3) \frac{1}{2}} {(x^{4})}^{\frac{1}{2}}}$

Step 15.6.2.2

Cancel the common factor.

$\frac{9 x^{8} z^{3}}{y^{4} \cdot 8 y^{2 \cdot 3 \frac{1}{2}} {(x^{4})}^{\frac{1}{2}}}$

Step 15.6.2.3

Rewrite the expression.

$\frac{9 x^{8} z^{3}}{y^{4} \cdot 8 y^{3} {(x^{4})}^{\frac{1}{2}}}$

Step 15.7

Multiply the exponents in ${(x^{4})}^{\frac{1}{2}}$ .

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Step 15.7.1

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

$\frac{9 x^{8} z^{3}}{y^{4} \cdot 8 y^{3} x^{4 (\frac{1}{2})}}$

Step 15.7.2

Cancel the common factor of $2$ .

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Step 15.7.2.1

Factor $2$ out of $4$ .

$\frac{9 x^{8} z^{3}}{y^{4} \cdot 8 y^{3} x^{2 (2) \frac{1}{2}}}$

Step 15.7.2.2

Cancel the common factor.

$\frac{9 x^{8} z^{3}}{y^{4} \cdot 8 y^{3} x^{2 \cdot 2 \frac{1}{2}}}$

Step 15.7.2.3

Rewrite the expression.

$\frac{9 x^{8} z^{3}}{y^{4} \cdot 8 y^{3} x^{2}}$

Step 15.8

Multiply $y^{4}$ by $y^{3}$ by adding the exponents.

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Step 15.8.1

Move $y^{3}$ .

$\frac{9 x^{8} z^{3}}{y^{3} y^{4} \cdot 8 x^{2}}$

Step 15.8.2

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

$\frac{9 x^{8} z^{3}}{y^{3 + 4} \cdot 8 x^{2}}$

Step 15.8.3

Add $3$ and $4$ .

$\frac{9 x^{8} z^{3}}{y^{7} \cdot 8 x^{2}}$

Step 16

Reduce the expression by cancelling the common factors.

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Step 16.1

Cancel the common factor of $x^{8}$ and $x^{2}$ .

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Step 16.1.1

Factor $x^{2}$ out of $9 x^{8} z^{3}$ .

$\frac{x^{2} (9 x^{6} z^{3})}{y^{7} \cdot 8 x^{2}}$

Step 16.1.2

Cancel the common factors.

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Step 16.1.2.1

Factor $x^{2}$ out of $y^{7} \cdot 8 x^{2}$ .

$\frac{x^{2} (9 x^{6} z^{3})}{x^{2} (y^{7} \cdot 8)}$

Step 16.1.2.2

Cancel the common factor.

$\frac{x^{2} (9 x^{6} z^{3})}{x^{2} (y^{7} \cdot 8)}$

Step 16.1.2.3

Rewrite the expression.

$\frac{9 x^{6} z^{3}}{y^{7} \cdot 8}$

Step 16.2

Move $8$ to the left of $y^{7}$ .

$\frac{9 x^{6} z^{3}}{8 y^{7}}$