Algebra Examples

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

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 $\frac{3 m^{2} n^{3}}{x y^{2}}$ .

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

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

Step 2

Simplify terms.

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Step 2.1

Combine.

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

Step 2.2

Cancel the common factor of $x^{3}$ and $x^{2}$ .

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Step 2.2.1

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

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

Step 2.2.2

Cancel the common factors.

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Step 2.2.2.1

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

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

Step 2.2.2.2

Cancel the common factor.

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

Step 2.2.2.3

Rewrite the expression.

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

Step 3

Simplify the numerator.

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

Multiply $2$ by $2$ .

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

Multiply $3$ by $2$ .

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

Step 4

Reduce the expression by cancelling the common factors.

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Step 4.1

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

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Step 4.1.1

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

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

Step 4.1.2

Multiply $2$ by $2$ .

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

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

Cancel the common factors.

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Step 4.2.2.1

Factor $m$ out of $y^{4} \cdot 2 m n^{2}$ .

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

Step 4.2.2.2

Cancel the common factor.

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

Step 4.2.2.3

Rewrite the expression.

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

Step 4.3

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

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Step 4.3.1

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

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

Step 4.3.2

Cancel the common factors.

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Step 4.3.2.1

Factor $n^{2}$ out of $y^{4} \cdot 2 n^{2}$ .

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

Step 4.3.2.2

Cancel the common factor.

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

Step 4.3.2.3

Rewrite the expression.

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

Step 4.4

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

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Step 4.4.1

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

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

Step 4.4.2

Cancel the common factors.

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Step 4.4.2.1

Factor $y$ out of $y^{4} \cdot 2$ .

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

Step 4.4.2.2

Cancel the common factor.

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

Step 4.4.2.3

Rewrite the expression.

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

Step 4.5

Move $2$ to the left of $y^{3}$ .

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