Basic Math Examples

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

Step 1

Multiply ${(y + 2)}^{5}$ by ${(y + 2)}^{3}$ by adding the exponents.

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Step 1.1

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

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

Step 1.2

Add $5$ and $3$ .

${(y + 2)}^{8}$

Step 2

Use the Binomial Theorem.

$y^{8} + 8 y^{7} \cdot 2 + 28 y^{6} \cdot 2^{2} + 56 y^{5} \cdot 2^{3} + 70 y^{4} \cdot 2^{4} + 56 y^{3} \cdot 2^{5} + 28 y^{2} \cdot 2^{6} + 8 y \cdot 2^{7} + 2^{8}$

Step 3

Simplify each term.

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Step 3.1

Multiply $2$ by $8$ .

$y^{8} + 16 y^{7} + 28 y^{6} \cdot 2^{2} + 56 y^{5} \cdot 2^{3} + 70 y^{4} \cdot 2^{4} + 56 y^{3} \cdot 2^{5} + 28 y^{2} \cdot 2^{6} + 8 y \cdot 2^{7} + 2^{8}$

Step 3.2

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

$y^{8} + 16 y^{7} + 28 y^{6} \cdot 4 + 56 y^{5} \cdot 2^{3} + 70 y^{4} \cdot 2^{4} + 56 y^{3} \cdot 2^{5} + 28 y^{2} \cdot 2^{6} + 8 y \cdot 2^{7} + 2^{8}$

Step 3.3

Multiply $4$ by $28$ .

$y^{8} + 16 y^{7} + 112 y^{6} + 56 y^{5} \cdot 2^{3} + 70 y^{4} \cdot 2^{4} + 56 y^{3} \cdot 2^{5} + 28 y^{2} \cdot 2^{6} + 8 y \cdot 2^{7} + 2^{8}$

Step 3.4

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

$y^{8} + 16 y^{7} + 112 y^{6} + 56 y^{5} \cdot 8 + 70 y^{4} \cdot 2^{4} + 56 y^{3} \cdot 2^{5} + 28 y^{2} \cdot 2^{6} + 8 y \cdot 2^{7} + 2^{8}$

Step 3.5

Multiply $8$ by $56$ .

$y^{8} + 16 y^{7} + 112 y^{6} + 448 y^{5} + 70 y^{4} \cdot 2^{4} + 56 y^{3} \cdot 2^{5} + 28 y^{2} \cdot 2^{6} + 8 y \cdot 2^{7} + 2^{8}$

Step 3.6

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

$y^{8} + 16 y^{7} + 112 y^{6} + 448 y^{5} + 70 y^{4} \cdot 16 + 56 y^{3} \cdot 2^{5} + 28 y^{2} \cdot 2^{6} + 8 y \cdot 2^{7} + 2^{8}$

Step 3.7

Multiply $16$ by $70$ .

$y^{8} + 16 y^{7} + 112 y^{6} + 448 y^{5} + 1120 y^{4} + 56 y^{3} \cdot 2^{5} + 28 y^{2} \cdot 2^{6} + 8 y \cdot 2^{7} + 2^{8}$

Step 3.8

Raise $2$ to the power of $5$ .

$y^{8} + 16 y^{7} + 112 y^{6} + 448 y^{5} + 1120 y^{4} + 56 y^{3} \cdot 32 + 28 y^{2} \cdot 2^{6} + 8 y \cdot 2^{7} + 2^{8}$

Step 3.9

Multiply $32$ by $56$ .

$y^{8} + 16 y^{7} + 112 y^{6} + 448 y^{5} + 1120 y^{4} + 1792 y^{3} + 28 y^{2} \cdot 2^{6} + 8 y \cdot 2^{7} + 2^{8}$

Step 3.10

Raise $2$ to the power of $6$ .

$y^{8} + 16 y^{7} + 112 y^{6} + 448 y^{5} + 1120 y^{4} + 1792 y^{3} + 28 y^{2} \cdot 64 + 8 y \cdot 2^{7} + 2^{8}$

Step 3.11

Multiply $64$ by $28$ .

$y^{8} + 16 y^{7} + 112 y^{6} + 448 y^{5} + 1120 y^{4} + 1792 y^{3} + 1792 y^{2} + 8 y \cdot 2^{7} + 2^{8}$

Step 3.12

Raise $2$ to the power of $7$ .

$y^{8} + 16 y^{7} + 112 y^{6} + 448 y^{5} + 1120 y^{4} + 1792 y^{3} + 1792 y^{2} + 8 y \cdot 128 + 2^{8}$

Step 3.13

Multiply $128$ by $8$ .

$y^{8} + 16 y^{7} + 112 y^{6} + 448 y^{5} + 1120 y^{4} + 1792 y^{3} + 1792 y^{2} + 1024 y + 2^{8}$

Step 3.14

Raise $2$ to the power of $8$ .

$y^{8} + 16 y^{7} + 112 y^{6} + 448 y^{5} + 1120 y^{4} + 1792 y^{3} + 1792 y^{2} + 1024 y + 256$