Basic Math Examples

$a^{y} (a^{y} - 4) \cdot (a^{y} + 1) - {(a^{y} - 1)}^{3}$

Step 1

Simplify each term.

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Step 1.1

Apply the distributive property.

$(a^{y} a^{y} + a^{y} \cdot - 4) \cdot (a^{y} + 1) - {(a^{y} - 1)}^{3}$

Step 1.2

Multiply $a^{y}$ by $a^{y}$ by adding the exponents.

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Step 1.2.1

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

$(a^{y + y} + a^{y} \cdot - 4) \cdot (a^{y} + 1) - {(a^{y} - 1)}^{3}$

Step 1.2.2

Add $y$ and $y$ .

$(a^{2 y} + a^{y} \cdot - 4) \cdot (a^{y} + 1) - {(a^{y} - 1)}^{3}$

Step 1.3

Move $- 4$ to the left of $a^{y}$ .

$(a^{2 y} - 4 \cdot a^{y}) \cdot (a^{y} + 1) - {(a^{y} - 1)}^{3}$

Step 1.4

Expand $(a^{2 y} - 4 a^{y}) (a^{y} + 1)$ using the FOIL Method.

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Step 1.4.1

Apply the distributive property.

$a^{2 y} (a^{y} + 1) - 4 a^{y} (a^{y} + 1) - {(a^{y} - 1)}^{3}$

Step 1.4.2

Apply the distributive property.

$a^{2 y} a^{y} + a^{2 y} \cdot 1 - 4 a^{y} (a^{y} + 1) - {(a^{y} - 1)}^{3}$

Step 1.4.3

Apply the distributive property.

$a^{2 y} a^{y} + a^{2 y} \cdot 1 - 4 a^{y} a^{y} - 4 a^{y} \cdot 1 - {(a^{y} - 1)}^{3}$

Step 1.5

Simplify and combine like terms.

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Step 1.5.1

Simplify each term.

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Step 1.5.1.1

Multiply $a^{2 y}$ by $a^{y}$ by adding the exponents.

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Step 1.5.1.1.1

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

$a^{2 y + y} + a^{2 y} \cdot 1 - 4 a^{y} a^{y} - 4 a^{y} \cdot 1 - {(a^{y} - 1)}^{3}$

Step 1.5.1.1.2

Add $2 y$ and $y$ .

$a^{3 y} + a^{2 y} \cdot 1 - 4 a^{y} a^{y} - 4 a^{y} \cdot 1 - {(a^{y} - 1)}^{3}$

Step 1.5.1.2

Multiply $a^{2 y}$ by $1$ .

$a^{3 y} + a^{2 y} - 4 a^{y} a^{y} - 4 a^{y} \cdot 1 - {(a^{y} - 1)}^{3}$

Step 1.5.1.3

Multiply $a^{y}$ by $a^{y}$ by adding the exponents.

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Step 1.5.1.3.1

Move $a^{y}$ .

$a^{3 y} + a^{2 y} - 4 (a^{y} a^{y}) - 4 a^{y} \cdot 1 - {(a^{y} - 1)}^{3}$

Step 1.5.1.3.2

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

$a^{3 y} + a^{2 y} - 4 a^{y + y} - 4 a^{y} \cdot 1 - {(a^{y} - 1)}^{3}$

Step 1.5.1.3.3

Add $y$ and $y$ .

$a^{3 y} + a^{2 y} - 4 a^{2 y} - 4 a^{y} \cdot 1 - {(a^{y} - 1)}^{3}$

Step 1.5.1.4

Multiply $- 4$ by $1$ .

$a^{3 y} + a^{2 y} - 4 a^{2 y} - 4 a^{y} - {(a^{y} - 1)}^{3}$

Step 1.5.2

Subtract $4 a^{2 y}$ from $a^{2 y}$ .

$a^{3 y} - 3 a^{2 y} - 4 a^{y} - {(a^{y} - 1)}^{3}$

Step 1.6

Use the Binomial Theorem.

$a^{3 y} - 3 a^{2 y} - 4 a^{y} - ({(a^{y})}^{3} + 3 {(a^{y})}^{2} \cdot - 1 + 3 a^{y} {(- 1)}^{2} + {(- 1)}^{3})$

Step 1.7

Simplify each term.

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Step 1.7.1

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

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Step 1.7.1.1

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

$a^{3 y} - 3 a^{2 y} - 4 a^{y} - (a^{y \cdot 3} + 3 {(a^{y})}^{2} \cdot - 1 + 3 a^{y} {(- 1)}^{2} + {(- 1)}^{3})$

Step 1.7.1.2

Move $3$ to the left of $y$ .

$a^{3 y} - 3 a^{2 y} - 4 a^{y} - (a^{3 y} + 3 {(a^{y})}^{2} \cdot - 1 + 3 a^{y} {(- 1)}^{2} + {(- 1)}^{3})$

Step 1.7.2

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

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Step 1.7.2.1

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

$a^{3 y} - 3 a^{2 y} - 4 a^{y} - (a^{3 y} + 3 a^{y \cdot 2} \cdot - 1 + 3 a^{y} {(- 1)}^{2} + {(- 1)}^{3})$

Step 1.7.2.2

Move $2$ to the left of $y$ .

$a^{3 y} - 3 a^{2 y} - 4 a^{y} - (a^{3 y} + 3 a^{2 y} \cdot - 1 + 3 a^{y} {(- 1)}^{2} + {(- 1)}^{3})$

Step 1.7.3

Multiply $- 1$ by $3$ .

$a^{3 y} - 3 a^{2 y} - 4 a^{y} - (a^{3 y} - 3 a^{2 y} + 3 a^{y} {(- 1)}^{2} + {(- 1)}^{3})$

Step 1.7.4

Raise $- 1$ to the power of $2$ .

$a^{3 y} - 3 a^{2 y} - 4 a^{y} - (a^{3 y} - 3 a^{2 y} + 3 a^{y} \cdot 1 + {(- 1)}^{3})$

Step 1.7.5

Multiply $3$ by $1$ .

$a^{3 y} - 3 a^{2 y} - 4 a^{y} - (a^{3 y} - 3 a^{2 y} + 3 a^{y} + {(- 1)}^{3})$

Step 1.7.6

Raise $- 1$ to the power of $3$ .

$a^{3 y} - 3 a^{2 y} - 4 a^{y} - (a^{3 y} - 3 a^{2 y} + 3 a^{y} - 1)$

Step 1.8

Apply the distributive property.

$a^{3 y} - 3 a^{2 y} - 4 a^{y} - a^{3 y} - (- 3 a^{2 y}) - (3 a^{y}) - - 1$

Step 1.9

Simplify.

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Step 1.9.1

Multiply $- 3$ by $- 1$ .

$a^{3 y} - 3 a^{2 y} - 4 a^{y} - a^{3 y} + 3 a^{2 y} - (3 a^{y}) - - 1$

Step 1.9.2

Multiply $3$ by $- 1$ .

$a^{3 y} - 3 a^{2 y} - 4 a^{y} - a^{3 y} + 3 a^{2 y} - 3 a^{y} - - 1$

Step 1.9.3

Multiply $- 1$ by $- 1$ .

$a^{3 y} - 3 a^{2 y} - 4 a^{y} - a^{3 y} + 3 a^{2 y} - 3 a^{y} + 1$

Step 2

Simplify by adding terms.

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Step 2.1

Combine the opposite terms in $a^{3 y} - 3 a^{2 y} - 4 a^{y} - a^{3 y} + 3 a^{2 y} - 3 a^{y} + 1$ .

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Step 2.1.1

Subtract $a^{3 y}$ from $a^{3 y}$ .

$- 3 a^{2 y} - 4 a^{y} + 0 + 3 a^{2 y} - 3 a^{y} + 1$

Step 2.1.2

Add $- 3 a^{2 y} - 4 a^{y}$ and $0$ .

$- 3 a^{2 y} - 4 a^{y} + 3 a^{2 y} - 3 a^{y} + 1$

Step 2.1.3

Add $- 3 a^{2 y}$ and $3 a^{2 y}$ .

$- 4 a^{y} + 0 - 3 a^{y} + 1$

Step 2.1.4

Add $- 4 a^{y}$ and $0$ .

$- 4 a^{y} - 3 a^{y} + 1$

Step 2.2

Subtract $3 a^{y}$ from $- 4 a^{y}$ .

$- 7 a^{y} + 1$