Basic Math Examples

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

Step 1

Simplify each term.

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Step 1.1

Use the Binomial Theorem.

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

Step 1.2

Simplify each term.

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Step 1.2.1

Multiply $2$ by $3$ .

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

Step 1.2.2

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

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

Step 1.2.3

Multiply $4$ by $3$ .

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

Step 1.2.4

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

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

Step 1.3

Apply the distributive property.

$y^{4} y^{3} + y^{4} (6 y^{2}) + y^{4} (12 y) + y^{4} \cdot 8 + y^{5} {(y + 2)}^{4}$

Step 1.4

Simplify.

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Step 1.4.1

Multiply $y^{4}$ by $y^{3}$ by adding the exponents.

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Step 1.4.1.1

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

$y^{4 + 3} + y^{4} (6 y^{2}) + y^{4} (12 y) + y^{4} \cdot 8 + y^{5} {(y + 2)}^{4}$

Step 1.4.1.2

Add $4$ and $3$ .

$y^{7} + y^{4} (6 y^{2}) + y^{4} (12 y) + y^{4} \cdot 8 + y^{5} {(y + 2)}^{4}$

Step 1.4.2

Rewrite using the commutative property of multiplication.

$y^{7} + 6 y^{4} y^{2} + y^{4} (12 y) + y^{4} \cdot 8 + y^{5} {(y + 2)}^{4}$

Step 1.4.3

Rewrite using the commutative property of multiplication.

$y^{7} + 6 y^{4} y^{2} + 12 y^{4} y + y^{4} \cdot 8 + y^{5} {(y + 2)}^{4}$

Step 1.4.4

Move $8$ to the left of $y^{4}$ .

$y^{7} + 6 y^{4} y^{2} + 12 y^{4} y + 8 \cdot y^{4} + y^{5} {(y + 2)}^{4}$

Step 1.5

Simplify each term.

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Step 1.5.1

Multiply $y^{4}$ by $y^{2}$ by adding the exponents.

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Step 1.5.1.1

Move $y^{2}$ .

$y^{7} + 6 (y^{2} y^{4}) + 12 y^{4} y + 8 \cdot y^{4} + y^{5} {(y + 2)}^{4}$

Step 1.5.1.2

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

$y^{7} + 6 y^{2 + 4} + 12 y^{4} y + 8 \cdot y^{4} + y^{5} {(y + 2)}^{4}$

Step 1.5.1.3

Add $2$ and $4$ .

$y^{7} + 6 y^{6} + 12 y^{4} y + 8 \cdot y^{4} + y^{5} {(y + 2)}^{4}$

Step 1.5.2

Multiply $y^{4}$ by $y$ by adding the exponents.

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Step 1.5.2.1

Move $y$ .

$y^{7} + 6 y^{6} + 12 (y \cdot y^{4}) + 8 \cdot y^{4} + y^{5} {(y + 2)}^{4}$

Step 1.5.2.2

Multiply $y$ by $y^{4}$ .

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Step 1.5.2.2.1

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

$y^{7} + 6 y^{6} + 12 (y^{1} y^{4}) + 8 \cdot y^{4} + y^{5} {(y + 2)}^{4}$

Step 1.5.2.2.2

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

$y^{7} + 6 y^{6} + 12 y^{1 + 4} + 8 \cdot y^{4} + y^{5} {(y + 2)}^{4}$

Step 1.5.2.3

Add $1$ and $4$ .

$y^{7} + 6 y^{6} + 12 y^{5} + 8 \cdot y^{4} + y^{5} {(y + 2)}^{4}$

$y^{7} + 6 y^{6} + 12 y^{5} + 8 y^{4} + y^{5} {(y + 2)}^{4}$

Step 1.6

Use the Binomial Theorem.

$y^{7} + 6 y^{6} + 12 y^{5} + 8 y^{4} + y^{5} (y^{4} + 4 y^{3} \cdot 2 + 6 y^{2} \cdot 2^{2} + 4 y \cdot 2^{3} + 2^{4})$

Step 1.7

Simplify each term.

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Step 1.7.1

Multiply $2$ by $4$ .

$y^{7} + 6 y^{6} + 12 y^{5} + 8 y^{4} + y^{5} (y^{4} + 8 y^{3} + 6 y^{2} \cdot 2^{2} + 4 y \cdot 2^{3} + 2^{4})$

Step 1.7.2

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

$y^{7} + 6 y^{6} + 12 y^{5} + 8 y^{4} + y^{5} (y^{4} + 8 y^{3} + 6 y^{2} \cdot 4 + 4 y \cdot 2^{3} + 2^{4})$

Step 1.7.3

Multiply $4$ by $6$ .

$y^{7} + 6 y^{6} + 12 y^{5} + 8 y^{4} + y^{5} (y^{4} + 8 y^{3} + 24 y^{2} + 4 y \cdot 2^{3} + 2^{4})$

Step 1.7.4

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

$y^{7} + 6 y^{6} + 12 y^{5} + 8 y^{4} + y^{5} (y^{4} + 8 y^{3} + 24 y^{2} + 4 y \cdot 8 + 2^{4})$

Step 1.7.5

Multiply $8$ by $4$ .

$y^{7} + 6 y^{6} + 12 y^{5} + 8 y^{4} + y^{5} (y^{4} + 8 y^{3} + 24 y^{2} + 32 y + 2^{4})$

Step 1.7.6

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

$y^{7} + 6 y^{6} + 12 y^{5} + 8 y^{4} + y^{5} (y^{4} + 8 y^{3} + 24 y^{2} + 32 y + 16)$

Step 1.8

Apply the distributive property.

$y^{7} + 6 y^{6} + 12 y^{5} + 8 y^{4} + y^{5} y^{4} + y^{5} (8 y^{3}) + y^{5} (24 y^{2}) + y^{5} (32 y) + y^{5} \cdot 16$

Step 1.9

Simplify.

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Step 1.9.1

Multiply $y^{5}$ by $y^{4}$ by adding the exponents.

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Step 1.9.1.1

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

$y^{7} + 6 y^{6} + 12 y^{5} + 8 y^{4} + y^{5 + 4} + y^{5} (8 y^{3}) + y^{5} (24 y^{2}) + y^{5} (32 y) + y^{5} \cdot 16$

Step 1.9.1.2

Add $5$ and $4$ .

$y^{7} + 6 y^{6} + 12 y^{5} + 8 y^{4} + y^{9} + y^{5} (8 y^{3}) + y^{5} (24 y^{2}) + y^{5} (32 y) + y^{5} \cdot 16$

Step 1.9.2

Rewrite using the commutative property of multiplication.

$y^{7} + 6 y^{6} + 12 y^{5} + 8 y^{4} + y^{9} + 8 y^{5} y^{3} + y^{5} (24 y^{2}) + y^{5} (32 y) + y^{5} \cdot 16$

Step 1.9.3

Rewrite using the commutative property of multiplication.

$y^{7} + 6 y^{6} + 12 y^{5} + 8 y^{4} + y^{9} + 8 y^{5} y^{3} + 24 y^{5} y^{2} + y^{5} (32 y) + y^{5} \cdot 16$

Step 1.9.4

Rewrite using the commutative property of multiplication.

$y^{7} + 6 y^{6} + 12 y^{5} + 8 y^{4} + y^{9} + 8 y^{5} y^{3} + 24 y^{5} y^{2} + 32 y^{5} y + y^{5} \cdot 16$

Step 1.9.5

Move $16$ to the left of $y^{5}$ .

$y^{7} + 6 y^{6} + 12 y^{5} + 8 y^{4} + y^{9} + 8 y^{5} y^{3} + 24 y^{5} y^{2} + 32 y^{5} y + 16 \cdot y^{5}$

Step 1.10

Simplify each term.

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Step 1.10.1

Multiply $y^{5}$ by $y^{3}$ by adding the exponents.

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Step 1.10.1.1

Move $y^{3}$ .

$y^{7} + 6 y^{6} + 12 y^{5} + 8 y^{4} + y^{9} + 8 (y^{3} y^{5}) + 24 y^{5} y^{2} + 32 y^{5} y + 16 \cdot y^{5}$

Step 1.10.1.2

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

$y^{7} + 6 y^{6} + 12 y^{5} + 8 y^{4} + y^{9} + 8 y^{3 + 5} + 24 y^{5} y^{2} + 32 y^{5} y + 16 \cdot y^{5}$

Step 1.10.1.3

Add $3$ and $5$ .

$y^{7} + 6 y^{6} + 12 y^{5} + 8 y^{4} + y^{9} + 8 y^{8} + 24 y^{5} y^{2} + 32 y^{5} y + 16 \cdot y^{5}$

Step 1.10.2

Multiply $y^{5}$ by $y^{2}$ by adding the exponents.

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Step 1.10.2.1

Move $y^{2}$ .

$y^{7} + 6 y^{6} + 12 y^{5} + 8 y^{4} + y^{9} + 8 y^{8} + 24 (y^{2} y^{5}) + 32 y^{5} y + 16 \cdot y^{5}$

Step 1.10.2.2

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

$y^{7} + 6 y^{6} + 12 y^{5} + 8 y^{4} + y^{9} + 8 y^{8} + 24 y^{2 + 5} + 32 y^{5} y + 16 \cdot y^{5}$

Step 1.10.2.3

Add $2$ and $5$ .

$y^{7} + 6 y^{6} + 12 y^{5} + 8 y^{4} + y^{9} + 8 y^{8} + 24 y^{7} + 32 y^{5} y + 16 \cdot y^{5}$

Step 1.10.3

Multiply $y^{5}$ by $y$ by adding the exponents.

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Step 1.10.3.1

Move $y$ .

$y^{7} + 6 y^{6} + 12 y^{5} + 8 y^{4} + y^{9} + 8 y^{8} + 24 y^{7} + 32 (y \cdot y^{5}) + 16 \cdot y^{5}$

Step 1.10.3.2

Multiply $y$ by $y^{5}$ .

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Step 1.10.3.2.1

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

$y^{7} + 6 y^{6} + 12 y^{5} + 8 y^{4} + y^{9} + 8 y^{8} + 24 y^{7} + 32 (y^{1} y^{5}) + 16 \cdot y^{5}$

Step 1.10.3.2.2

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

$y^{7} + 6 y^{6} + 12 y^{5} + 8 y^{4} + y^{9} + 8 y^{8} + 24 y^{7} + 32 y^{1 + 5} + 16 \cdot y^{5}$

Step 1.10.3.3

Add $1$ and $5$ .

$y^{7} + 6 y^{6} + 12 y^{5} + 8 y^{4} + y^{9} + 8 y^{8} + 24 y^{7} + 32 y^{6} + 16 \cdot y^{5}$

$y^{7} + 6 y^{6} + 12 y^{5} + 8 y^{4} + y^{9} + 8 y^{8} + 24 y^{7} + 32 y^{6} + 16 y^{5}$

Step 2

Simplify by adding terms.

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Step 2.1

Add $y^{7}$ and $24 y^{7}$ .

$25 y^{7} + 6 y^{6} + 12 y^{5} + 8 y^{4} + y^{9} + 8 y^{8} + 32 y^{6} + 16 y^{5}$

Step 2.2

Add $6 y^{6}$ and $32 y^{6}$ .

$25 y^{7} + 38 y^{6} + 12 y^{5} + 8 y^{4} + y^{9} + 8 y^{8} + 16 y^{5}$

Step 2.3

Add $12 y^{5}$ and $16 y^{5}$ .

$25 y^{7} + 38 y^{6} + 8 y^{4} + y^{9} + 8 y^{8} + 28 y^{5}$

Step 2.4

Reorder.

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Step 2.4.1

Move $8 y^{4}$ .

$25 y^{7} + 38 y^{6} + y^{9} + 8 y^{8} + 28 y^{5} + 8 y^{4}$

Step 2.4.2

Move $38 y^{6}$ .

$25 y^{7} + y^{9} + 8 y^{8} + 38 y^{6} + 28 y^{5} + 8 y^{4}$

Step 2.4.3

Move $25 y^{7}$ .

$y^{9} + 8 y^{8} + 25 y^{7} + 38 y^{6} + 28 y^{5} + 8 y^{4}$