Algebra Examples

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

Step 1

Multiply $x^{- \frac{1}{2}}$ by $x^{\frac{1}{3}}$ by adding the exponents.

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Step 1.1

Move $x^{\frac{1}{3}}$ .

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

Step 1.2

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

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

Step 1.3

To write $\frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 1.4

To write $- \frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 1.5

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.5.1

Multiply $\frac{1}{3}$ by $\frac{2}{2}$ .

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

Step 1.5.2

Multiply $3$ by $2$ .

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

Step 1.5.3

Multiply $\frac{1}{2}$ by $\frac{3}{3}$ .

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

Step 1.5.4

Multiply $2$ by $3$ .

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

Step 1.6

Combine the numerators over the common denominator.

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

Step 1.7

Subtract $3$ from $2$ .

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

Step 1.8

Move the negative in front of the fraction.

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

Step 2

Multiply $x^{- \frac{1}{6}}$ by $x^{2}$ by adding the exponents.

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Step 2.1

Move $x^{2}$ .

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

Step 2.2

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

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

Step 2.3

To write $2$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

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

Step 2.4

Combine $2$ and $\frac{6}{6}$ .

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

Step 2.5

Combine the numerators over the common denominator.

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

Step 2.6

Simplify the numerator.

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Step 2.6.1

Multiply $2$ by $6$ .

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

Step 2.6.2

Subtract $1$ from $12$ .

${(x^{\frac{11}{6}} y^{- 1} z^{\frac{1}{3}} (y^{- \frac{1}{3}} z) (y^{- 3} z^{- \frac{1}{2}}))}^{6}$

Step 3

Multiply $y^{- 1}$ by $y^{- \frac{1}{3}}$ by adding the exponents.

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Step 3.1

Move $y^{- \frac{1}{3}}$ .

${(x^{\frac{11}{6}} (y^{- \frac{1}{3}} y^{- 1}) z^{\frac{1}{3}} z (y^{- 3} z^{- \frac{1}{2}}))}^{6}$

Step 3.2

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

${(x^{\frac{11}{6}} y^{- \frac{1}{3} - 1} z^{\frac{1}{3}} z (y^{- 3} z^{- \frac{1}{2}}))}^{6}$

Step 3.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

${(x^{\frac{11}{6}} y^{- \frac{1}{3} - 1 \cdot \frac{3}{3}} z^{\frac{1}{3}} z (y^{- 3} z^{- \frac{1}{2}}))}^{6}$

Step 3.4

Combine $- 1$ and $\frac{3}{3}$ .

${(x^{\frac{11}{6}} y^{- \frac{1}{3} + \frac{- 1 \cdot 3}{3}} z^{\frac{1}{3}} z (y^{- 3} z^{- \frac{1}{2}}))}^{6}$

Step 3.5

Combine the numerators over the common denominator.

${(x^{\frac{11}{6}} y^{\frac{- 1 - 1 \cdot 3}{3}} z^{\frac{1}{3}} z (y^{- 3} z^{- \frac{1}{2}}))}^{6}$

Step 3.6

Simplify the numerator.

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Step 3.6.1

Multiply $- 1$ by $3$ .

${(x^{\frac{11}{6}} y^{\frac{- 1 - 3}{3}} z^{\frac{1}{3}} z (y^{- 3} z^{- \frac{1}{2}}))}^{6}$

Step 3.6.2

Subtract $3$ from $- 1$ .

${(x^{\frac{11}{6}} y^{\frac{- 4}{3}} z^{\frac{1}{3}} z (y^{- 3} z^{- \frac{1}{2}}))}^{6}$

Step 3.7

Move the negative in front of the fraction.

${(x^{\frac{11}{6}} y^{- \frac{4}{3}} z^{\frac{1}{3}} z (y^{- 3} z^{- \frac{1}{2}}))}^{6}$

Step 4

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

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Step 4.1

Move $y^{- 3}$ .

${(x^{\frac{11}{6}} (y^{- 3} y^{- \frac{4}{3}}) z^{\frac{1}{3}} z \cdot z^{- \frac{1}{2}})}^{6}$

Step 4.2

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

${(x^{\frac{11}{6}} y^{- 3 - \frac{4}{3}} z^{\frac{1}{3}} z \cdot z^{- \frac{1}{2}})}^{6}$

Step 4.3

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

${(x^{\frac{11}{6}} y^{- 3 \cdot \frac{3}{3} - \frac{4}{3}} z^{\frac{1}{3}} z \cdot z^{- \frac{1}{2}})}^{6}$

Step 4.4

Combine $- 3$ and $\frac{3}{3}$ .

${(x^{\frac{11}{6}} y^{\frac{- 3 \cdot 3}{3} - \frac{4}{3}} z^{\frac{1}{3}} z \cdot z^{- \frac{1}{2}})}^{6}$

Step 4.5

Combine the numerators over the common denominator.

${(x^{\frac{11}{6}} y^{\frac{- 3 \cdot 3 - 4}{3}} z^{\frac{1}{3}} z \cdot z^{- \frac{1}{2}})}^{6}$

Step 4.6

Simplify the numerator.

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Step 4.6.1

Multiply $- 3$ by $3$ .

${(x^{\frac{11}{6}} y^{\frac{- 9 - 4}{3}} z^{\frac{1}{3}} z \cdot z^{- \frac{1}{2}})}^{6}$

Step 4.6.2

Subtract $4$ from $- 9$ .

${(x^{\frac{11}{6}} y^{\frac{- 13}{3}} z^{\frac{1}{3}} z \cdot z^{- \frac{1}{2}})}^{6}$

Step 4.7

Move the negative in front of the fraction.

${(x^{\frac{11}{6}} y^{- \frac{13}{3}} z^{\frac{1}{3}} z \cdot z^{- \frac{1}{2}})}^{6}$

Step 5

Multiply $z^{\frac{1}{3}}$ by $z$ by adding the exponents.

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Step 5.1

Move $z$ .

${(x^{\frac{11}{6}} y^{- \frac{13}{3}} (z \cdot z^{\frac{1}{3}}) z^{- \frac{1}{2}})}^{6}$

Step 5.2

Multiply $z$ by $z^{\frac{1}{3}}$ .

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Step 5.2.1

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

${(x^{\frac{11}{6}} y^{- \frac{13}{3}} (z^{1} z^{\frac{1}{3}}) z^{- \frac{1}{2}})}^{6}$

Step 5.2.2

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

${(x^{\frac{11}{6}} y^{- \frac{13}{3}} z^{1 + \frac{1}{3}} z^{- \frac{1}{2}})}^{6}$

Step 5.3

Write $1$ as a fraction with a common denominator.

${(x^{\frac{11}{6}} y^{- \frac{13}{3}} z^{\frac{3}{3} + \frac{1}{3}} z^{- \frac{1}{2}})}^{6}$

Step 5.4

Combine the numerators over the common denominator.

${(x^{\frac{11}{6}} y^{- \frac{13}{3}} z^{\frac{3 + 1}{3}} z^{- \frac{1}{2}})}^{6}$

Step 5.5

Add $3$ and $1$ .

${(x^{\frac{11}{6}} y^{- \frac{13}{3}} z^{\frac{4}{3}} z^{- \frac{1}{2}})}^{6}$

Step 6

Multiply $z^{\frac{4}{3}}$ by $z^{- \frac{1}{2}}$ by adding the exponents.

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Step 6.1

Move $z^{- \frac{1}{2}}$ .

${(x^{\frac{11}{6}} y^{- \frac{13}{3}} (z^{- \frac{1}{2}} z^{\frac{4}{3}}))}^{6}$

Step 6.2

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

${(x^{\frac{11}{6}} y^{- \frac{13}{3}} z^{- \frac{1}{2} + \frac{4}{3}})}^{6}$

Step 6.3

To write $- \frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

${(x^{\frac{11}{6}} y^{- \frac{13}{3}} z^{- \frac{1}{2} \cdot \frac{3}{3} + \frac{4}{3}})}^{6}$

Step 6.4

To write $\frac{4}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

${(x^{\frac{11}{6}} y^{- \frac{13}{3}} z^{- \frac{1}{2} \cdot \frac{3}{3} + \frac{4}{3} \cdot \frac{2}{2}})}^{6}$

Step 6.5

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.5.1

Multiply $\frac{1}{2}$ by $\frac{3}{3}$ .

${(x^{\frac{11}{6}} y^{- \frac{13}{3}} z^{- \frac{3}{2 \cdot 3} + \frac{4}{3} \cdot \frac{2}{2}})}^{6}$

Step 6.5.2

Multiply $2$ by $3$ .

${(x^{\frac{11}{6}} y^{- \frac{13}{3}} z^{- \frac{3}{6} + \frac{4}{3} \cdot \frac{2}{2}})}^{6}$

Step 6.5.3

Multiply $\frac{4}{3}$ by $\frac{2}{2}$ .

${(x^{\frac{11}{6}} y^{- \frac{13}{3}} z^{- \frac{3}{6} + \frac{4 \cdot 2}{3 \cdot 2}})}^{6}$

Step 6.5.4

Multiply $3$ by $2$ .

${(x^{\frac{11}{6}} y^{- \frac{13}{3}} z^{- \frac{3}{6} + \frac{4 \cdot 2}{6}})}^{6}$

Step 6.6

Combine the numerators over the common denominator.

${(x^{\frac{11}{6}} y^{- \frac{13}{3}} z^{\frac{- 3 + 4 \cdot 2}{6}})}^{6}$

Step 6.7

Simplify the numerator.

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Step 6.7.1

Multiply $4$ by $2$ .

${(x^{\frac{11}{6}} y^{- \frac{13}{3}} z^{\frac{- 3 + 8}{6}})}^{6}$

Step 6.7.2

Add $- 3$ and $8$ .

${(x^{\frac{11}{6}} y^{- \frac{13}{3}} z^{\frac{5}{6}})}^{6}$

Step 7

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

${(x^{\frac{11}{6}} \frac{1}{y^{\frac{13}{3}}} z^{\frac{5}{6}})}^{6}$

Step 8

Combine $x^{\frac{11}{6}}$ and $\frac{1}{y^{\frac{13}{3}}}$ .

${(\frac{x^{\frac{11}{6}}}{y^{\frac{13}{3}}} z^{\frac{5}{6}})}^{6}$

Step 9

Combine $\frac{x^{\frac{11}{6}}}{y^{\frac{13}{3}}}$ and $z^{\frac{5}{6}}$ .

${(\frac{x^{\frac{11}{6}} z^{\frac{5}{6}}}{y^{\frac{13}{3}}})}^{6}$

Step 10

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

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Step 10.1

Apply the product rule to $\frac{x^{\frac{11}{6}} z^{\frac{5}{6}}}{y^{\frac{13}{3}}}$ .

$\frac{{(x^{\frac{11}{6}} z^{\frac{5}{6}})}^{6}}{{(y^{\frac{13}{3}})}^{6}}$

Step 10.2

Apply the product rule to $x^{\frac{11}{6}} z^{\frac{5}{6}}$ .

$\frac{{(x^{\frac{11}{6}})}^{6} {(z^{\frac{5}{6}})}^{6}}{{(y^{\frac{13}{3}})}^{6}}$

Step 11

Simplify the numerator.

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Step 11.1

Multiply the exponents in ${(x^{\frac{11}{6}})}^{6}$ .

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Step 11.1.1

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

$\frac{x^{\frac{11}{6} \cdot 6} {(z^{\frac{5}{6}})}^{6}}{{(y^{\frac{13}{3}})}^{6}}$

Step 11.1.2

Cancel the common factor of $6$ .

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Step 11.1.2.1

Cancel the common factor.

$\frac{x^{\frac{11}{6} \cdot 6} {(z^{\frac{5}{6}})}^{6}}{{(y^{\frac{13}{3}})}^{6}}$

Step 11.1.2.2

Rewrite the expression.

$\frac{x^{11} {(z^{\frac{5}{6}})}^{6}}{{(y^{\frac{13}{3}})}^{6}}$

Step 11.2

Multiply the exponents in ${(z^{\frac{5}{6}})}^{6}$ .

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Step 11.2.1

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

$\frac{x^{11} z^{\frac{5}{6} \cdot 6}}{{(y^{\frac{13}{3}})}^{6}}$

Step 11.2.2

Cancel the common factor of $6$ .

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Step 11.2.2.1

Cancel the common factor.

$\frac{x^{11} z^{\frac{5}{6} \cdot 6}}{{(y^{\frac{13}{3}})}^{6}}$

Step 11.2.2.2

Rewrite the expression.

$\frac{x^{11} z^{5}}{{(y^{\frac{13}{3}})}^{6}}$

Step 12

Multiply the exponents in ${(y^{\frac{13}{3}})}^{6}$ .

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Step 12.1

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

$\frac{x^{11} z^{5}}{y^{\frac{13}{3} \cdot 6}}$

Step 12.2

Cancel the common factor of $3$ .

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Step 12.2.1

Factor $3$ out of $6$ .

$\frac{x^{11} z^{5}}{y^{\frac{13}{3} \cdot (3 (2))}}$

Step 12.2.2

Cancel the common factor.

$\frac{x^{11} z^{5}}{y^{\frac{13}{3} \cdot (3 \cdot 2)}}$

Step 12.2.3

Rewrite the expression.

$\frac{x^{11} z^{5}}{y^{13 \cdot 2}}$

Step 12.3

Multiply $13$ by $2$ .

$\frac{x^{11} z^{5}}{y^{26}}$