Basic Math Examples

$m^{6} - l^{6}$

Step 1

Rewrite $m^{6}$ as ${(m^{2})}^{3}$ .

${(m^{2})}^{3} - l^{6}$

Step 2

Rewrite $l^{6}$ as ${(l^{2})}^{3}$ .

${(m^{2})}^{3} - {(l^{2})}^{3}$

Step 3

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = m^{2}$ and $b = l^{2}$ .

$(m^{2} - l^{2}) ({(m^{2})}^{2} + m^{2} l^{2} + {(l^{2})}^{2})$

Step 4

Simplify.

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Step 4.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = m$ and $b = l$ .

$(m + l) (m - l) ({(m^{2})}^{2} + m^{2} l^{2} + {(l^{2})}^{2})$

Step 4.2

Multiply the exponents in ${(m^{2})}^{2}$ .

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Step 4.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$(m + l) (m - l) (m^{2 \cdot 2} + m^{2} l^{2} + {(l^{2})}^{2})$

Step 4.2.2

Multiply $2$ by $2$ .

$(m + l) (m - l) (m^{4} + m^{2} l^{2} + {(l^{2})}^{2})$

Step 4.3

Multiply the exponents in ${(l^{2})}^{2}$ .

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Step 4.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$(m + l) (m - l) (m^{4} + m^{2} l^{2} + l^{2 \cdot 2})$

Step 4.3.2

Multiply $2$ by $2$ .

$(m + l) (m - l) (m^{4} + m^{2} l^{2} + l^{4})$

Step 4.4

Factor.

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Step 4.4.1

Rewrite $m^{4} + m^{2} l^{2} + l^{4}$ in a factored form.

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Step 4.4.1.1

Rewrite the middle term.

$(m + l) (m - l) (m^{4} + 2 m^{2} l^{2} - m^{2} l^{2} + l^{4})$

Step 4.4.1.2

Rearrange terms.

$(m + l) (m - l) (m^{4} + 2 m^{2} l^{2} + l^{4} - m^{2} l^{2})$

Step 4.4.1.3

Factor first three terms by perfect square rule.

$(m + l) (m - l) ({(m^{2} + l^{2})}^{2} - m^{2} l^{2})$

Step 4.4.1.4

Rewrite $m^{2} l^{2}$ as ${(m l)}^{2}$ .

$(m + l) (m - l) ({(m^{2} + l^{2})}^{2} - {(m l)}^{2})$

Step 4.4.1.5

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = m^{2} + l^{2}$ and $b = m l$ .

$(m + l) (m - l) ((m^{2} + l^{2} + m l) (m^{2} + l^{2} - (m l)))$

Step 4.4.1.6

Remove parentheses.

$(m + l) (m - l) ((m^{2} + l^{2} + m l) (m^{2} + l^{2} - m l))$

Step 4.4.2

Remove unnecessary parentheses.

$(m + l) (m - l) (m^{2} + l^{2} + m l) (m^{2} + l^{2} - m l)$