Algebra Examples

${(\frac{3 m^{2} n^{3}}{x y^{2}})}^{2} {(\frac{x^{3} y z^{2}}{2 m n^{2}})}^{3}$

Step 1

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

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Step 1.1

Apply the product rule to $\frac{3 m^{2} n^{3}}{x y^{2}}$ .

$\frac{{(3 m^{2} n^{3})}^{2}}{{(x y^{2})}^{2}} {(\frac{x^{3} y z^{2}}{2 m n^{2}})}^{3}$

Step 1.2

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

$\frac{{(3 m^{2})}^{2} {(n^{3})}^{2}}{{(x y^{2})}^{2}} {(\frac{x^{3} y z^{2}}{2 m n^{2}})}^{3}$

Step 1.3

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

$\frac{3^{2} {(m^{2})}^{2} {(n^{3})}^{2}}{{(x y^{2})}^{2}} {(\frac{x^{3} y z^{2}}{2 m n^{2}})}^{3}$

Step 1.4

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

$\frac{3^{2} {(m^{2})}^{2} {(n^{3})}^{2}}{x^{2} {(y^{2})}^{2}} {(\frac{x^{3} y z^{2}}{2 m n^{2}})}^{3}$

Step 2

Simplify the numerator.

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Step 2.1

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

$\frac{9 {(m^{2})}^{2} {(n^{3})}^{2}}{x^{2} {(y^{2})}^{2}} {(\frac{x^{3} y z^{2}}{2 m n^{2}})}^{3}$

Step 2.2

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

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Step 2.2.1

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

$\frac{9 m^{2 \cdot 2} {(n^{3})}^{2}}{x^{2} {(y^{2})}^{2}} {(\frac{x^{3} y z^{2}}{2 m n^{2}})}^{3}$

Step 2.2.2

Multiply $2$ by $2$ .

$\frac{9 m^{4} {(n^{3})}^{2}}{x^{2} {(y^{2})}^{2}} {(\frac{x^{3} y z^{2}}{2 m n^{2}})}^{3}$

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

Multiply $3$ by $2$ .

$\frac{9 m^{4} n^{6}}{x^{2} {(y^{2})}^{2}} {(\frac{x^{3} y z^{2}}{2 m n^{2}})}^{3}$

Step 3

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

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Step 3.1

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

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

Step 3.2

Multiply $2$ by $2$ .

$\frac{9 m^{4} n^{6}}{x^{2} y^{4}} {(\frac{x^{3} y z^{2}}{2 m n^{2}})}^{3}$

Step 4

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

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Step 4.1

Apply the product rule to $\frac{x^{3} y z^{2}}{2 m n^{2}}$ .

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

Apply the product rule to $2 m n^{2}$ .

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

Step 4.5

Apply the product rule to $2 m$ .

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

Step 5

Simplify terms.

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Step 5.1

Combine.

$\frac{9 m^{4} n^{6} ({(x^{3})}^{3} y^{3} {(z^{2})}^{3})}{x^{2} y^{4} (2^{3} m^{3} {(n^{2})}^{3})}$

Step 5.2

Cancel the common factor of $m^{4}$ and $m^{3}$ .

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Step 5.2.1

Factor $m^{3}$ out of $9 m^{4} n^{6} ({(x^{3})}^{3} y^{3} {(z^{2})}^{3})$ .

$\frac{m^{3} (9 m n^{6} ({(x^{3})}^{3} y^{3} {(z^{2})}^{3}))}{x^{2} y^{4} (2^{3} m^{3} {(n^{2})}^{3})}$

Step 5.2.2

Cancel the common factors.

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Step 5.2.2.1

Factor $m^{3}$ out of $x^{2} y^{4} (2^{3} m^{3} {(n^{2})}^{3})$ .

$\frac{m^{3} (9 m n^{6} ({(x^{3})}^{3} y^{3} {(z^{2})}^{3}))}{m^{3} (x^{2} y^{4} (2^{3} {(n^{2})}^{3}))}$

Step 5.2.2.2

Cancel the common factor.

$\frac{m^{3} (9 m n^{6} ({(x^{3})}^{3} y^{3} {(z^{2})}^{3}))}{m^{3} (x^{2} y^{4} (2^{3} {(n^{2})}^{3}))}$

Step 5.2.2.3

Rewrite the expression.

$\frac{9 m n^{6} ({(x^{3})}^{3} y^{3} {(z^{2})}^{3})}{x^{2} y^{4} (2^{3} {(n^{2})}^{3})}$

Step 5.3

Cancel the common factor of $y^{3}$ and $y^{4}$ .

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Step 5.3.1

Factor $y^{3}$ out of $9 m n^{6} ({(x^{3})}^{3} y^{3} {(z^{2})}^{3})$ .

$\frac{y^{3} (9 m n^{6} ({(x^{3})}^{3} {(z^{2})}^{3}))}{x^{2} y^{4} (2^{3} {(n^{2})}^{3})}$

Step 5.3.2

Cancel the common factors.

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Step 5.3.2.1

Factor $y^{3}$ out of $x^{2} y^{4} (2^{3} {(n^{2})}^{3})$ .

$\frac{y^{3} (9 m n^{6} ({(x^{3})}^{3} {(z^{2})}^{3}))}{y^{3} (x^{2} y (2^{3} {(n^{2})}^{3}))}$

Step 5.3.2.2

Cancel the common factor.

$\frac{y^{3} (9 m n^{6} ({(x^{3})}^{3} {(z^{2})}^{3}))}{y^{3} (x^{2} y (2^{3} {(n^{2})}^{3}))}$

Step 5.3.2.3

Rewrite the expression.

$\frac{9 m n^{6} ({(x^{3})}^{3} {(z^{2})}^{3})}{x^{2} y (2^{3} {(n^{2})}^{3})}$

Step 6

Simplify the numerator.

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Step 6.1

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

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Step 6.1.1

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

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

Step 6.1.2

Multiply $3$ by $3$ .

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

Step 6.2

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

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Step 6.2.1

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

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

Step 6.2.2

Multiply $2$ by $3$ .

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

Step 7

Simplify the denominator.

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Step 7.1

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

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

Step 7.2

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

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Step 7.2.1

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

$\frac{9 m n^{6} x^{9} z^{6}}{x^{2} y \cdot 8 n^{2 \cdot 3}}$

Step 7.2.2

Multiply $2$ by $3$ .

$\frac{9 m n^{6} x^{9} z^{6}}{x^{2} y \cdot 8 n^{6}}$

Step 8

Reduce the expression by cancelling the common factors.

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Step 8.1

Cancel the common factor of $n^{6}$ .

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Step 8.1.1

Cancel the common factor.

$\frac{9 m n^{6} x^{9} z^{6}}{x^{2} y \cdot 8 n^{6}}$

Step 8.1.2

Rewrite the expression.

$\frac{9 m x^{9} z^{6}}{x^{2} y \cdot 8}$

Step 8.2

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

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Step 8.2.1

Factor $x^{2}$ out of $9 m x^{9} z^{6}$ .

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

Step 8.2.2

Cancel the common factors.

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Step 8.2.2.1

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

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

Step 8.2.2.2

Cancel the common factor.

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

Step 8.2.2.3

Rewrite the expression.

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

Step 8.3

Move $8$ to the left of $y$ .

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