Algebra Examples

$r^{27} - s^{30}$

Step 1

Rewrite $r^{27}$ as ${(r^{9})}^{3}$ .

${(r^{9})}^{3} - s^{30}$

Step 2

Rewrite $s^{30}$ as ${(s^{10})}^{3}$ .

${(r^{9})}^{3} - {(s^{10})}^{3}$

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 $a = r^{9}$ and $b = s^{10}$ .

$(r^{9} - s^{10}) ({(r^{9})}^{2} + r^{9} s^{10} + {(s^{10})}^{2})$

Step 4

Simplify.

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Step 4.1

Multiply the exponents in ${(r^{9})}^{2}$ .

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Step 4.1.1

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

$(r^{9} - s^{10}) (r^{9 \cdot 2} + r^{9} s^{10} + {(s^{10})}^{2})$

Step 4.1.2

Multiply $9$ by $2$ .

$(r^{9} - s^{10}) (r^{18} + r^{9} s^{10} + {(s^{10})}^{2})$

Step 4.2

Multiply the exponents in ${(s^{10})}^{2}$ .

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Step 4.2.1

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

$(r^{9} - s^{10}) (r^{18} + r^{9} s^{10} + s^{10 \cdot 2})$

Step 4.2.2

Multiply $10$ by $2$ .

$(r^{9} - s^{10}) (r^{18} + r^{9} s^{10} + s^{20})$