Algebra Examples

${(a^{2} + b^{2} + c^{2})}^{2}$

Step 1

Rewrite ${(a^{2} + b^{2} + c^{2})}^{2}$ as $(a^{2} + b^{2} + c^{2}) (a^{2} + b^{2} + c^{2})$ .

$(a^{2} + b^{2} + c^{2}) (a^{2} + b^{2} + c^{2})$

Step 2

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

$a^{2} a^{2} + a^{2} b^{2} + a^{2} c^{2} + b^{2} a^{2} + b^{2} b^{2} + b^{2} c^{2} + c^{2} a^{2} + c^{2} b^{2} + c^{2} c^{2}$

Step 3

Simplify each term.

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Step 3.1

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

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Step 3.1.1

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

$a^{2 + 2} + a^{2} b^{2} + a^{2} c^{2} + b^{2} a^{2} + b^{2} b^{2} + b^{2} c^{2} + c^{2} a^{2} + c^{2} b^{2} + c^{2} c^{2}$

Step 3.1.2

Add $2$ and $2$ .

$a^{4} + a^{2} b^{2} + a^{2} c^{2} + b^{2} a^{2} + b^{2} b^{2} + b^{2} c^{2} + c^{2} a^{2} + c^{2} b^{2} + c^{2} c^{2}$

Step 3.2

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

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Step 3.2.1

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

$a^{4} + a^{2} b^{2} + a^{2} c^{2} + b^{2} a^{2} + b^{2 + 2} + b^{2} c^{2} + c^{2} a^{2} + c^{2} b^{2} + c^{2} c^{2}$

Step 3.2.2

Add $2$ and $2$ .

$a^{4} + a^{2} b^{2} + a^{2} c^{2} + b^{2} a^{2} + b^{4} + b^{2} c^{2} + c^{2} a^{2} + c^{2} b^{2} + c^{2} c^{2}$

Step 3.3

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

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Step 3.3.1

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

$a^{4} + a^{2} b^{2} + a^{2} c^{2} + b^{2} a^{2} + b^{4} + b^{2} c^{2} + c^{2} a^{2} + c^{2} b^{2} + c^{2 + 2}$

Step 3.3.2

Add $2$ and $2$ .

$a^{4} + a^{2} b^{2} + a^{2} c^{2} + b^{2} a^{2} + b^{4} + b^{2} c^{2} + c^{2} a^{2} + c^{2} b^{2} + c^{4}$

Step 4

Add $a^{2} b^{2}$ and $b^{2} a^{2}$ .

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Step 4.1

Reorder $b^{2}$ and $a^{2}$ .

$a^{4} + a^{2} b^{2} + a^{2} b^{2} + a^{2} c^{2} + b^{4} + b^{2} c^{2} + c^{2} a^{2} + c^{2} b^{2} + c^{4}$

Step 4.2

Add $a^{2} b^{2}$ and $a^{2} b^{2}$ .

$a^{4} + 2 a^{2} b^{2} + a^{2} c^{2} + b^{4} + b^{2} c^{2} + c^{2} a^{2} + c^{2} b^{2} + c^{4}$

Step 5

Add $a^{2} c^{2}$ and $c^{2} a^{2}$ .

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Step 5.1

Reorder $c^{2}$ and $a^{2}$ .

$a^{4} + 2 a^{2} b^{2} + b^{4} + b^{2} c^{2} + a^{2} c^{2} + a^{2} c^{2} + c^{2} b^{2} + c^{4}$

Step 5.2

Add $a^{2} c^{2}$ and $a^{2} c^{2}$ .

$a^{4} + 2 a^{2} b^{2} + b^{4} + b^{2} c^{2} + 2 a^{2} c^{2} + c^{2} b^{2} + c^{4}$

Step 6

Add $b^{2} c^{2}$ and $c^{2} b^{2}$ .

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Step 6.1

Reorder $c^{2}$ and $b^{2}$ .

$a^{4} + 2 a^{2} b^{2} + b^{4} + b^{2} c^{2} + b^{2} c^{2} + 2 a^{2} c^{2} + c^{4}$

Step 6.2

Add $b^{2} c^{2}$ and $b^{2} c^{2}$ .

$a^{4} + 2 a^{2} b^{2} + b^{4} + 2 b^{2} c^{2} + 2 a^{2} c^{2} + c^{4}$