Algebra Examples

$5 {(x^{2} - 2 y)}^{3}$

Step 1

Use the Binomial Theorem.

$5 ({(x^{2})}^{3} + 3 {(x^{2})}^{2} (- 2 y) + 3 x^{2} {(- 2 y)}^{2} + {(- 2 y)}^{3})$

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

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

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Step 2.1.1.1

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

$5 (x^{2 \cdot 3} + 3 {(x^{2})}^{2} (- 2 y) + 3 x^{2} {(- 2 y)}^{2} + {(- 2 y)}^{3})$

Step 2.1.1.2

Multiply $2$ by $3$ .

$5 (x^{6} + 3 {(x^{2})}^{2} (- 2 y) + 3 x^{2} {(- 2 y)}^{2} + {(- 2 y)}^{3})$

Step 2.1.2

Rewrite using the commutative property of multiplication.

$5 (x^{6} + 3 \cdot - 2 {(x^{2})}^{2} y + 3 x^{2} {(- 2 y)}^{2} + {(- 2 y)}^{3})$

Step 2.1.3

Multiply $3$ by $- 2$ .

$5 (x^{6} - 6 {(x^{2})}^{2} y + 3 x^{2} {(- 2 y)}^{2} + {(- 2 y)}^{3})$

Step 2.1.4

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

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Step 2.1.4.1

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

$5 (x^{6} - 6 x^{2 \cdot 2} y + 3 x^{2} {(- 2 y)}^{2} + {(- 2 y)}^{3})$

Step 2.1.4.2

Multiply $2$ by $2$ .

$5 (x^{6} - 6 x^{4} y + 3 x^{2} {(- 2 y)}^{2} + {(- 2 y)}^{3})$

Step 2.1.5

Apply the product rule to $- 2 y$ .

$5 (x^{6} - 6 x^{4} y + 3 x^{2} ({(- 2)}^{2} y^{2}) + {(- 2 y)}^{3})$

Step 2.1.6

Rewrite using the commutative property of multiplication.

$5 (x^{6} - 6 x^{4} y + 3 \cdot {(- 2)}^{2} x^{2} y^{2} + {(- 2 y)}^{3})$

Step 2.1.7

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

$5 (x^{6} - 6 x^{4} y + 3 \cdot 4 x^{2} y^{2} + {(- 2 y)}^{3})$

Step 2.1.8

Multiply $3$ by $4$ .

$5 (x^{6} - 6 x^{4} y + 12 x^{2} y^{2} + {(- 2 y)}^{3})$

Step 2.1.9

Apply the product rule to $- 2 y$ .

$5 (x^{6} - 6 x^{4} y + 12 x^{2} y^{2} + {(- 2)}^{3} y^{3})$

Step 2.1.10

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

$5 (x^{6} - 6 x^{4} y + 12 x^{2} y^{2} - 8 y^{3})$

Step 2.2

Apply the distributive property.

$5 x^{6} + 5 (- 6 x^{4} y) + 5 (12 x^{2} y^{2}) + 5 (- 8 y^{3})$

Step 3

Simplify.

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Step 3.1

Multiply $- 6$ by $5$ .

$5 x^{6} - 30 (x^{4} y) + 5 (12 x^{2} y^{2}) + 5 (- 8 y^{3})$

Step 3.2

Multiply $12$ by $5$ .

$5 x^{6} - 30 (x^{4} y) + 60 (x^{2} y^{2}) + 5 (- 8 y^{3})$

Step 3.3

Multiply $- 8$ by $5$ .

$5 x^{6} - 30 (x^{4} y) + 60 (x^{2} y^{2}) - 40 y^{3}$

Step 4

Remove parentheses.

$5 x^{6} - 30 x^{4} y + 60 x^{2} y^{2} - 40 y^{3}$