Basic Math Examples

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

Step 1

Simplify each term.

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Step 1.1

Use the Binomial Theorem.

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

Step 1.2

Simplify each term.

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Step 1.2.1

Multiply $3$ by $3$ .

$y^{3} + 9 y^{2} + 3 y \cdot 3^{2} + 3^{3} - {(y - 3)}^{2}$

Step 1.2.2

Multiply $3$ by $3^{2}$ by adding the exponents.

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Step 1.2.2.1

Move $3^{2}$ .

$y^{3} + 9 y^{2} + 3^{2} \cdot 3 y + 3^{3} - {(y - 3)}^{2}$

Step 1.2.2.2

Multiply $3^{2}$ by $3$ .

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Step 1.2.2.2.1

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

$y^{3} + 9 y^{2} + 3^{2} \cdot 3^{1} y + 3^{3} - {(y - 3)}^{2}$

Step 1.2.2.2.2

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

$y^{3} + 9 y^{2} + 3^{2 + 1} y + 3^{3} - {(y - 3)}^{2}$

Step 1.2.2.3

Add $2$ and $1$ .

$y^{3} + 9 y^{2} + 3^{3} y + 3^{3} - {(y - 3)}^{2}$

Step 1.2.3

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

$y^{3} + 9 y^{2} + 27 y + 3^{3} - {(y - 3)}^{2}$

Step 1.2.4

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

$y^{3} + 9 y^{2} + 27 y + 27 - {(y - 3)}^{2}$

Step 1.3

Rewrite ${(y - 3)}^{2}$ as $(y - 3) (y - 3)$ .

$y^{3} + 9 y^{2} + 27 y + 27 - ((y - 3) (y - 3))$

Step 1.4

Expand $(y - 3) (y - 3)$ using the FOIL Method.

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Step 1.4.1

Apply the distributive property.

$y^{3} + 9 y^{2} + 27 y + 27 - (y (y - 3) - 3 (y - 3))$

Step 1.4.2

Apply the distributive property.

$y^{3} + 9 y^{2} + 27 y + 27 - (y \cdot y + y \cdot - 3 - 3 (y - 3))$

Step 1.4.3

Apply the distributive property.

$y^{3} + 9 y^{2} + 27 y + 27 - (y \cdot y + y \cdot - 3 - 3 y - 3 \cdot - 3)$

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 $y$ by $y$ .

$y^{3} + 9 y^{2} + 27 y + 27 - (y^{2} + y \cdot - 3 - 3 y - 3 \cdot - 3)$

Step 1.5.1.2

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

$y^{3} + 9 y^{2} + 27 y + 27 - (y^{2} - 3 \cdot y - 3 y - 3 \cdot - 3)$

Step 1.5.1.3

Multiply $- 3$ by $- 3$ .

$y^{3} + 9 y^{2} + 27 y + 27 - (y^{2} - 3 y - 3 y + 9)$

Step 1.5.2

Subtract $3 y$ from $- 3 y$ .

$y^{3} + 9 y^{2} + 27 y + 27 - (y^{2} - 6 y + 9)$

Step 1.6

Apply the distributive property.

$y^{3} + 9 y^{2} + 27 y + 27 - y^{2} - (- 6 y) - 1 \cdot 9$

Step 1.7

Simplify.

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Step 1.7.1

Multiply $- 6$ by $- 1$ .

$y^{3} + 9 y^{2} + 27 y + 27 - y^{2} + 6 y - 1 \cdot 9$

Step 1.7.2

Multiply $- 1$ by $9$ .

$y^{3} + 9 y^{2} + 27 y + 27 - y^{2} + 6 y - 9$

Step 2

Simplify by adding terms.

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Step 2.1

Subtract $y^{2}$ from $9 y^{2}$ .

$y^{3} + 8 y^{2} + 27 y + 27 + 6 y - 9$

Step 2.2

Add $27 y$ and $6 y$ .

$y^{3} + 8 y^{2} + 33 y + 27 - 9$

Step 2.3

Subtract $9$ from $27$ .

$y^{3} + 8 y^{2} + 33 y + 18$