Algebra Examples

$t^{12} - r^{3} s^{15}$

Step 1

Rewrite $t^{12}$ as ${(t^{4})}^{3}$ .

${(t^{4})}^{3} - r^{3} s^{15}$

Step 2

Rewrite $r^{3} s^{15}$ as ${(r s^{5})}^{3}$ .

${(t^{4})}^{3} - {(r s^{5})}^{3}$

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 $a = t^{4}$ and $b = r s^{5}$ .

$(t^{4} - (r s^{5})) ({(t^{4})}^{2} + t^{4} (r s^{5}) + {(r s^{5})}^{2})$

Step 4

Simplify.

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Step 4.1

Remove parentheses.

$(t^{4} - r s^{5}) ({(t^{4})}^{2} + t^{4} (r s^{5}) + {(r s^{5})}^{2})$

Step 4.2

Multiply the exponents in ${(t^{4})}^{2}$ .

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Step 4.2.1

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

$(t^{4} - r s^{5}) (t^{4 \cdot 2} + t^{4} (r s^{5}) + {(r s^{5})}^{2})$

Step 4.2.2

Multiply $4$ by $2$ .

$(t^{4} - r s^{5}) (t^{8} + t^{4} (r s^{5}) + {(r s^{5})}^{2})$

Step 4.3

Remove parentheses.

$(t^{4} - r s^{5}) (t^{8} + t^{4} r s^{5} + {(r s^{5})}^{2})$

Step 4.4

Apply the product rule to $r s^{5}$ .

$(t^{4} - r s^{5}) (t^{8} + t^{4} r s^{5} + r^{2} {(s^{5})}^{2})$

Step 4.5

Multiply the exponents in ${(s^{5})}^{2}$ .

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Step 4.5.1

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

$(t^{4} - r s^{5}) (t^{8} + t^{4} r s^{5} + r^{2} s^{5 \cdot 2})$

Step 4.5.2

Multiply $5$ by $2$ .

$(t^{4} - r s^{5}) (t^{8} + t^{4} r s^{5} + r^{2} s^{10})$