Algebra Examples

${(- 3 x y^{2})}^{3} \cdot {(- \frac{2}{3} x^{2} y^{3})}^{4}$

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 $- 3 x y^{2}$ .

${(- 3 x)}^{3} {(y^{2})}^{3} \cdot {(- \frac{2}{3} \cdot (x^{2} y^{3}))}^{4}$

Step 1.2

Apply the product rule to $- 3 x$ .

${(- 3)}^{3} x^{3} {(y^{2})}^{3} \cdot {(- \frac{2}{3} \cdot (x^{2} y^{3}))}^{4}$

Step 2

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

$- 27 x^{3} {(y^{2})}^{3} \cdot {(- \frac{2}{3} \cdot (x^{2} y^{3}))}^{4}$

Step 3

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

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Step 3.1

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

$- 27 x^{3} y^{2 \cdot 3} \cdot {(- \frac{2}{3} \cdot (x^{2} y^{3}))}^{4}$

Step 3.2

Multiply $2$ by $3$ .

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

Step 4

Multiply $- \frac{2}{3} (x^{2} y^{3})$ .

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Step 4.1

Combine $x^{2}$ and $\frac{2}{3}$ .

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

Step 4.2

Combine $y^{3}$ and $\frac{x^{2} \cdot 2}{3}$ .

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

Step 5

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

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

Step 6

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

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

Step 6.4

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

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

Step 7

Simplify the expression.

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Step 7.1

Raise $- 1$ to the power of $4$ .

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

Step 7.2

Multiply $\frac{2^{4} {(y^{3})}^{4} {(x^{2})}^{4}}{3^{4}}$ by $1$ .

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

Step 8

Simplify the numerator.

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Step 8.1

Raise $2$ to the power of $4$ .

$- 27 x^{3} y^{6} \cdot \frac{16 {(y^{3})}^{4} {(x^{2})}^{4}}{3^{4}}$

Step 8.2

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

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Step 8.2.1

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

$- 27 x^{3} y^{6} \cdot \frac{16 y^{3 \cdot 4} {(x^{2})}^{4}}{3^{4}}$

Step 8.2.2

Multiply $3$ by $4$ .

$- 27 x^{3} y^{6} \cdot \frac{16 y^{12} {(x^{2})}^{4}}{3^{4}}$

Step 8.3

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

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Step 8.3.1

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

$- 27 x^{3} y^{6} \cdot \frac{16 y^{12} x^{2 \cdot 4}}{3^{4}}$

Step 8.3.2

Multiply $2$ by $4$ .

$- 27 x^{3} y^{6} \cdot \frac{16 y^{12} x^{8}}{3^{4}}$

Step 9

Simplify terms.

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Step 9.1

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

$- 27 x^{3} y^{6} \cdot \frac{16 y^{12} x^{8}}{81}$

Step 9.2

Cancel the common factor of $27$ .

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Step 9.2.1

Factor $27$ out of $- 27 x^{3} y^{6}$ .

$27 (- x^{3} y^{6}) \cdot \frac{16 y^{12} x^{8}}{81}$

Step 9.2.2

Factor $27$ out of $81$ .

$27 (- x^{3} y^{6}) \cdot \frac{16 y^{12} x^{8}}{27 (3)}$

Step 9.2.3

Cancel the common factor.

$27 (- x^{3} y^{6}) \cdot \frac{16 y^{12} x^{8}}{27 \cdot 3}$

Step 9.2.4

Rewrite the expression.

$- x^{3} y^{6} \cdot \frac{16 y^{12} x^{8}}{3}$

Step 9.3

Combine $x^{3}$ and $\frac{16 y^{12} x^{8}}{3}$ .

$- 1 y^{6} \cdot \frac{x^{3} (16 y^{12} x^{8})}{3}$

Step 10

Multiply $x^{3}$ by $x^{8}$ by adding the exponents.

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Step 10.1

Move $x^{8}$ .

$- 1 y^{6} \cdot \frac{x^{8} x^{3} (16 y^{12})}{3}$

Step 10.2

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

$- 1 y^{6} \cdot \frac{x^{8 + 3} (16 y^{12})}{3}$

Step 10.3

Add $8$ and $3$ .

$- 1 y^{6} \cdot \frac{x^{11} (16 y^{12})}{3}$

Step 11

Combine $\frac{x^{11} (16 y^{12})}{3}$ and $y^{6}$ .

$- 1 \frac{x^{11} (16 y^{12}) y^{6}}{3}$

Step 12

Multiply $y^{12}$ by $y^{6}$ by adding the exponents.

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Step 12.1

Move $y^{6}$ .

$- 1 \frac{x^{11} (16 (y^{6} y^{12}))}{3}$

Step 12.2

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

$- 1 \frac{x^{11} (16 y^{6 + 12})}{3}$

Step 12.3

Add $6$ and $12$ .

$- 1 \frac{x^{11} (16 y^{18})}{3}$

Step 13

Move $16$ to the left of $x^{11}$ .

$- 1 \frac{16 \cdot x^{11} y^{18}}{3}$

Step 14

Rewrite $- 1 \frac{16 x^{11} y^{18}}{3}$ as $- \frac{16 x^{11} y^{18}}{3}$ .

$- \frac{16 x^{11} y^{18}}{3}$