Algebra Examples

$(a b^{2} - 5) (a^{2} b^{4} + 5 a b^{2} + 25)$

Step 1

Expand $(a b^{2} - 5) (a^{2} b^{4} + 5 a b^{2} + 25)$ by multiplying each term in the first expression by each term in the second expression.

$a b^{2} (a^{2} b^{4}) + a b^{2} (5 a b^{2}) + a b^{2} \cdot 25 - 5 (a^{2} b^{4}) - 5 (5 a b^{2}) - 5 \cdot 25$

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

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

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Step 2.1.1.1

Move $a^{2}$ .

$a^{2} a b^{2} b^{4} + a b^{2} (5 a b^{2}) + a b^{2} \cdot 25 - 5 (a^{2} b^{4}) - 5 (5 a b^{2}) - 5 \cdot 25$

Step 2.1.1.2

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

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Step 2.1.1.2.1

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

$a^{2} a^{1} b^{2} b^{4} + a b^{2} (5 a b^{2}) + a b^{2} \cdot 25 - 5 (a^{2} b^{4}) - 5 (5 a b^{2}) - 5 \cdot 25$

Step 2.1.1.2.2

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

$a^{2 + 1} b^{2} b^{4} + a b^{2} (5 a b^{2}) + a b^{2} \cdot 25 - 5 (a^{2} b^{4}) - 5 (5 a b^{2}) - 5 \cdot 25$

Step 2.1.1.3

Add $2$ and $1$ .

$a^{3} b^{2} b^{4} + a b^{2} (5 a b^{2}) + a b^{2} \cdot 25 - 5 (a^{2} b^{4}) - 5 (5 a b^{2}) - 5 \cdot 25$

Step 2.1.2

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

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Step 2.1.2.1

Move $b^{4}$ .

$a^{3} (b^{4} b^{2}) + a b^{2} (5 a b^{2}) + a b^{2} \cdot 25 - 5 (a^{2} b^{4}) - 5 (5 a b^{2}) - 5 \cdot 25$

Step 2.1.2.2

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

$a^{3} b^{4 + 2} + a b^{2} (5 a b^{2}) + a b^{2} \cdot 25 - 5 (a^{2} b^{4}) - 5 (5 a b^{2}) - 5 \cdot 25$

Step 2.1.2.3

Add $4$ and $2$ .

$a^{3} b^{6} + a b^{2} (5 a b^{2}) + a b^{2} \cdot 25 - 5 (a^{2} b^{4}) - 5 (5 a b^{2}) - 5 \cdot 25$

Step 2.1.3

Rewrite using the commutative property of multiplication.

$a^{3} b^{6} + 5 a b^{2} (a b^{2}) + a b^{2} \cdot 25 - 5 (a^{2} b^{4}) - 5 (5 a b^{2}) - 5 \cdot 25$

Step 2.1.4

Multiply $a$ by $a$ by adding the exponents.

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Step 2.1.4.1

Move $a$ .

$a^{3} b^{6} + 5 (a \cdot a) b^{2} b^{2} + a b^{2} \cdot 25 - 5 (a^{2} b^{4}) - 5 (5 a b^{2}) - 5 \cdot 25$

Step 2.1.4.2

Multiply $a$ by $a$ .

$a^{3} b^{6} + 5 a^{2} b^{2} b^{2} + a b^{2} \cdot 25 - 5 (a^{2} b^{4}) - 5 (5 a b^{2}) - 5 \cdot 25$

Step 2.1.5

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

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Step 2.1.5.1

Move $b^{2}$ .

$a^{3} b^{6} + 5 a^{2} (b^{2} b^{2}) + a b^{2} \cdot 25 - 5 (a^{2} b^{4}) - 5 (5 a b^{2}) - 5 \cdot 25$

Step 2.1.5.2

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

$a^{3} b^{6} + 5 a^{2} b^{2 + 2} + a b^{2} \cdot 25 - 5 (a^{2} b^{4}) - 5 (5 a b^{2}) - 5 \cdot 25$

Step 2.1.5.3

Add $2$ and $2$ .

$a^{3} b^{6} + 5 a^{2} b^{4} + a b^{2} \cdot 25 - 5 (a^{2} b^{4}) - 5 (5 a b^{2}) - 5 \cdot 25$

Step 2.1.6

Move $25$ to the left of $a b^{2}$ .

$a^{3} b^{6} + 5 a^{2} b^{4} + 25 \cdot (a b^{2}) - 5 (a^{2} b^{4}) - 5 (5 a b^{2}) - 5 \cdot 25$

Step 2.1.7

Multiply $5$ by $- 5$ .

$a^{3} b^{6} + 5 a^{2} b^{4} + 25 a b^{2} - 5 a^{2} b^{4} - 25 (a b^{2}) - 5 \cdot 25$

Step 2.1.8

Multiply $- 5$ by $25$ .

$a^{3} b^{6} + 5 a^{2} b^{4} + 25 a b^{2} - 5 a^{2} b^{4} - 25 a b^{2} - 125$

Step 2.2

Combine the opposite terms in $a^{3} b^{6} + 5 a^{2} b^{4} + 25 a b^{2} - 5 a^{2} b^{4} - 25 a b^{2} - 125$ .

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Step 2.2.1

Subtract $5 a^{2} b^{4}$ from $5 a^{2} b^{4}$ .

$a^{3} b^{6} + 25 a b^{2} + 0 - 25 a b^{2} - 125$

Step 2.2.2

Add $a^{3} b^{6} + 25 a b^{2}$ and $0$ .

$a^{3} b^{6} + 25 a b^{2} - 25 a b^{2} - 125$

Step 2.2.3

Subtract $25 a b^{2}$ from $25 a b^{2}$ .

$a^{3} b^{6} + 0 - 125$

Step 2.2.4

Add $a^{3} b^{6}$ and $0$ .

$a^{3} b^{6} - 125$