Algebra Examples

${(\frac{4 x^{3} y^{4}}{3 m^{2} n^{3}})}^{4} {(\frac{9 m^{5} n}{2 x^{2} y^{5}})}^{2}$

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

$\frac{{(4 x^{3} y^{4})}^{4}}{{(3 m^{2} n^{3})}^{4}} {(\frac{9 m^{5} n}{2 x^{2} y^{5}})}^{2}$

Step 1.2

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

$\frac{{(4 x^{3})}^{4} {(y^{4})}^{4}}{{(3 m^{2} n^{3})}^{4}} {(\frac{9 m^{5} n}{2 x^{2} y^{5}})}^{2}$

Step 1.3

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

$\frac{4^{4} {(x^{3})}^{4} {(y^{4})}^{4}}{{(3 m^{2} n^{3})}^{4}} {(\frac{9 m^{5} n}{2 x^{2} y^{5}})}^{2}$

Step 1.4

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

$\frac{4^{4} {(x^{3})}^{4} {(y^{4})}^{4}}{{(3 m^{2})}^{4} {(n^{3})}^{4}} {(\frac{9 m^{5} n}{2 x^{2} y^{5}})}^{2}$

Step 1.5

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

$\frac{4^{4} {(x^{3})}^{4} {(y^{4})}^{4}}{3^{4} {(m^{2})}^{4} {(n^{3})}^{4}} {(\frac{9 m^{5} n}{2 x^{2} y^{5}})}^{2}$

Step 2

Simplify the numerator.

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Step 2.1

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

$\frac{256 {(x^{3})}^{4} {(y^{4})}^{4}}{3^{4} {(m^{2})}^{4} {(n^{3})}^{4}} {(\frac{9 m^{5} n}{2 x^{2} y^{5}})}^{2}$

Step 2.2

Multiply the exponents in ${(x^{3})}^{4}$ .

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Step 2.2.1

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

$\frac{256 x^{3 \cdot 4} {(y^{4})}^{4}}{3^{4} {(m^{2})}^{4} {(n^{3})}^{4}} {(\frac{9 m^{5} n}{2 x^{2} y^{5}})}^{2}$

Step 2.2.2

Multiply $3$ by $4$ .

$\frac{256 x^{12} {(y^{4})}^{4}}{3^{4} {(m^{2})}^{4} {(n^{3})}^{4}} {(\frac{9 m^{5} n}{2 x^{2} y^{5}})}^{2}$

Step 2.3

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

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Step 2.3.1

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

$\frac{256 x^{12} y^{4 \cdot 4}}{3^{4} {(m^{2})}^{4} {(n^{3})}^{4}} {(\frac{9 m^{5} n}{2 x^{2} y^{5}})}^{2}$

Step 2.3.2

Multiply $4$ by $4$ .

$\frac{256 x^{12} y^{16}}{3^{4} {(m^{2})}^{4} {(n^{3})}^{4}} {(\frac{9 m^{5} n}{2 x^{2} y^{5}})}^{2}$

Step 3

Simplify the denominator.

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Step 3.1

Raise $3$ to the power of $4$ .

$\frac{256 x^{12} y^{16}}{81 {(m^{2})}^{4} {(n^{3})}^{4}} {(\frac{9 m^{5} n}{2 x^{2} y^{5}})}^{2}$

Step 3.2

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

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Step 3.2.1

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

$\frac{256 x^{12} y^{16}}{81 m^{2 \cdot 4} {(n^{3})}^{4}} {(\frac{9 m^{5} n}{2 x^{2} y^{5}})}^{2}$

Step 3.2.2

Multiply $2$ by $4$ .

$\frac{256 x^{12} y^{16}}{81 m^{8} {(n^{3})}^{4}} {(\frac{9 m^{5} n}{2 x^{2} y^{5}})}^{2}$

Step 3.3

Multiply the exponents in ${(n^{3})}^{4}$ .

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Step 3.3.1

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

$\frac{256 x^{12} y^{16}}{81 m^{8} n^{3 \cdot 4}} {(\frac{9 m^{5} n}{2 x^{2} y^{5}})}^{2}$

Step 3.3.2

Multiply $3$ by $4$ .

$\frac{256 x^{12} y^{16}}{81 m^{8} n^{12}} {(\frac{9 m^{5} n}{2 x^{2} y^{5}})}^{2}$

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 $\frac{9 m^{5} n}{2 x^{2} y^{5}}$ .

$\frac{256 x^{12} y^{16}}{81 m^{8} n^{12}} \cdot \frac{{(9 m^{5} n)}^{2}}{{(2 x^{2} y^{5})}^{2}}$

Step 4.2

Apply the product rule to $9 m^{5} n$ .

$\frac{256 x^{12} y^{16}}{81 m^{8} n^{12}} \cdot \frac{{(9 m^{5})}^{2} n^{2}}{{(2 x^{2} y^{5})}^{2}}$

Step 4.3

Apply the product rule to $9 m^{5}$ .

$\frac{256 x^{12} y^{16}}{81 m^{8} n^{12}} \cdot \frac{9^{2} {(m^{5})}^{2} n^{2}}{{(2 x^{2} y^{5})}^{2}}$

Step 4.4

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

$\frac{256 x^{12} y^{16}}{81 m^{8} n^{12}} \cdot \frac{9^{2} {(m^{5})}^{2} n^{2}}{{(2 x^{2})}^{2} {(y^{5})}^{2}}$

Step 4.5

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

$\frac{256 x^{12} y^{16}}{81 m^{8} n^{12}} \cdot \frac{9^{2} {(m^{5})}^{2} n^{2}}{2^{2} {(x^{2})}^{2} {(y^{5})}^{2}}$

Step 5

Simplify terms.

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Step 5.1

Combine.

$\frac{256 x^{12} y^{16} (9^{2} {(m^{5})}^{2} n^{2})}{81 m^{8} n^{12} (2^{2} {(x^{2})}^{2} {(y^{5})}^{2})}$

Step 5.2

Cancel the common factor of $n^{2}$ and $n^{12}$ .

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Step 5.2.1

Factor $n^{2}$ out of $256 x^{12} y^{16} (9^{2} {(m^{5})}^{2} n^{2})$ .

$\frac{n^{2} (256 x^{12} y^{16} (9^{2} {(m^{5})}^{2}))}{81 m^{8} n^{12} (2^{2} {(x^{2})}^{2} {(y^{5})}^{2})}$

Step 5.2.2

Cancel the common factors.

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Step 5.2.2.1

Factor $n^{2}$ out of $81 m^{8} n^{12} (2^{2} {(x^{2})}^{2} {(y^{5})}^{2})$ .

$\frac{n^{2} (256 x^{12} y^{16} (9^{2} {(m^{5})}^{2}))}{n^{2} (81 m^{8} n^{10} (2^{2} {(x^{2})}^{2} {(y^{5})}^{2}))}$

Step 5.2.2.2

Cancel the common factor.

$\frac{n^{2} (256 x^{12} y^{16} (9^{2} {(m^{5})}^{2}))}{n^{2} (81 m^{8} n^{10} (2^{2} {(x^{2})}^{2} {(y^{5})}^{2}))}$

Step 5.2.2.3

Rewrite the expression.

$\frac{256 x^{12} y^{16} (9^{2} {(m^{5})}^{2})}{81 m^{8} n^{10} (2^{2} {(x^{2})}^{2} {(y^{5})}^{2})}$

Step 6

Simplify the numerator.

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Step 6.1

Raise $9$ to the power of $2$ .

$\frac{256 x^{12} y^{16} \cdot 81 {(m^{5})}^{2}}{81 m^{8} n^{10} \cdot 2^{2} {(x^{2})}^{2} {(y^{5})}^{2}}$

Step 6.2

Multiply the exponents in ${(m^{5})}^{2}$ .

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Step 6.2.1

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

$\frac{256 x^{12} y^{16} \cdot 81 m^{5 \cdot 2}}{81 m^{8} n^{10} \cdot 2^{2} {(x^{2})}^{2} {(y^{5})}^{2}}$

Step 6.2.2

Multiply $5$ by $2$ .

$\frac{256 x^{12} y^{16} \cdot 81 m^{10}}{81 m^{8} n^{10} \cdot 2^{2} {(x^{2})}^{2} {(y^{5})}^{2}}$

Step 6.3

Multiply $81$ by $256$ .

$\frac{20736 x^{12} y^{16} m^{10}}{81 m^{8} n^{10} \cdot 2^{2} {(x^{2})}^{2} {(y^{5})}^{2}}$

Step 7

Simplify the denominator.

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Step 7.1

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

$\frac{20736 x^{12} y^{16} m^{10}}{81 m^{8} n^{10} \cdot 4 {(x^{2})}^{2} {(y^{5})}^{2}}$

Step 7.2

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

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Step 7.2.1

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

$\frac{20736 x^{12} y^{16} m^{10}}{81 m^{8} n^{10} \cdot 4 x^{2 \cdot 2} {(y^{5})}^{2}}$

Step 7.2.2

Multiply $2$ by $2$ .

$\frac{20736 x^{12} y^{16} m^{10}}{81 m^{8} n^{10} \cdot 4 x^{4} {(y^{5})}^{2}}$

Step 7.3

Multiply the exponents in ${(y^{5})}^{2}$ .

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Step 7.3.1

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

$\frac{20736 x^{12} y^{16} m^{10}}{81 m^{8} n^{10} \cdot 4 x^{4} y^{5 \cdot 2}}$

Step 7.3.2

Multiply $5$ by $2$ .

$\frac{20736 x^{12} y^{16} m^{10}}{81 m^{8} n^{10} \cdot 4 x^{4} y^{10}}$

Step 7.4

Multiply $4$ by $81$ .

$\frac{20736 x^{12} y^{16} m^{10}}{324 m^{8} n^{10} x^{4} y^{10}}$

Step 8

Reduce the expression by cancelling the common factors.

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Step 8.1

Cancel the common factor of $20736$ and $324$ .

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Step 8.1.1

Factor $324$ out of $20736 x^{12} y^{16} m^{10}$ .

$\frac{324 (64 x^{12} y^{16} m^{10})}{324 m^{8} n^{10} x^{4} y^{10}}$

Step 8.1.2

Cancel the common factors.

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Step 8.1.2.1

Factor $324$ out of $324 m^{8} n^{10} x^{4} y^{10}$ .

$\frac{324 (64 x^{12} y^{16} m^{10})}{324 (m^{8} n^{10} x^{4} y^{10})}$

Step 8.1.2.2

Cancel the common factor.

$\frac{324 (64 x^{12} y^{16} m^{10})}{324 (m^{8} n^{10} x^{4} y^{10})}$

Step 8.1.2.3

Rewrite the expression.

$\frac{64 x^{12} y^{16} m^{10}}{m^{8} n^{10} x^{4} y^{10}}$

Step 8.2

Cancel the common factor of $x^{12}$ and $x^{4}$ .

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Step 8.2.1

Factor $x^{4}$ out of $64 x^{12} y^{16} m^{10}$ .

$\frac{x^{4} (64 x^{8} y^{16} m^{10})}{m^{8} n^{10} x^{4} y^{10}}$

Step 8.2.2

Cancel the common factors.

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Step 8.2.2.1

Factor $x^{4}$ out of $m^{8} n^{10} x^{4} y^{10}$ .

$\frac{x^{4} (64 x^{8} y^{16} m^{10})}{x^{4} (m^{8} n^{10} y^{10})}$

Step 8.2.2.2

Cancel the common factor.

$\frac{x^{4} (64 x^{8} y^{16} m^{10})}{x^{4} (m^{8} n^{10} y^{10})}$

Step 8.2.2.3

Rewrite the expression.

$\frac{64 x^{8} y^{16} m^{10}}{m^{8} n^{10} y^{10}}$

Step 8.3

Cancel the common factor of $y^{16}$ and $y^{10}$ .

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Step 8.3.1

Factor $y^{10}$ out of $64 x^{8} y^{16} m^{10}$ .

$\frac{y^{10} (64 x^{8} y^{6} m^{10})}{m^{8} n^{10} y^{10}}$

Step 8.3.2

Cancel the common factors.

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Step 8.3.2.1

Factor $y^{10}$ out of $m^{8} n^{10} y^{10}$ .

$\frac{y^{10} (64 x^{8} y^{6} m^{10})}{y^{10} (m^{8} n^{10})}$

Step 8.3.2.2

Cancel the common factor.

$\frac{y^{10} (64 x^{8} y^{6} m^{10})}{y^{10} (m^{8} n^{10})}$

Step 8.3.2.3

Rewrite the expression.

$\frac{64 x^{8} y^{6} m^{10}}{m^{8} n^{10}}$

Step 8.4

Cancel the common factor of $m^{10}$ and $m^{8}$ .

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Step 8.4.1

Factor $m^{8}$ out of $64 x^{8} y^{6} m^{10}$ .

$\frac{m^{8} (64 x^{8} y^{6} m^{2})}{m^{8} n^{10}}$

Step 8.4.2

Cancel the common factors.

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Step 8.4.2.1

Factor $m^{8}$ out of $m^{8} n^{10}$ .

$\frac{m^{8} (64 x^{8} y^{6} m^{2})}{m^{8} (n^{10})}$

Step 8.4.2.2

Cancel the common factor.

$\frac{m^{8} (64 x^{8} y^{6} m^{2})}{m^{8} n^{10}}$

Step 8.4.2.3

Rewrite the expression.

$\frac{64 x^{8} y^{6} m^{2}}{n^{10}}$