Basic Math Examples

$125 m^{12} - n^{15}$

Step 1

Rewrite $125 m^{12}$ as ${(5 m^{4})}^{3}$ .

${(5 m^{4})}^{3} - n^{15}$

Step 2

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

${(5 m^{4})}^{3} - {(n^{5})}^{3}$

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 $a = 5 m^{4}$ and $b = n^{5}$ .

$(5 m^{4} - n^{5}) ({(5 m^{4})}^{2} + 5 m^{4} n^{5} + {(n^{5})}^{2})$

Step 4

Simplify.

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Step 4.1

Apply the product rule to $5 m^{4}$ .

$(5 m^{4} - n^{5}) (5^{2} {(m^{4})}^{2} + 5 m^{4} n^{5} + {(n^{5})}^{2})$

Step 4.2

Raise $5$ to the power of $2$ .

$(5 m^{4} - n^{5}) (25 {(m^{4})}^{2} + 5 m^{4} n^{5} + {(n^{5})}^{2})$

Step 4.3

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

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Step 4.3.1

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

$(5 m^{4} - n^{5}) (25 m^{4 \cdot 2} + 5 m^{4} n^{5} + {(n^{5})}^{2})$

Step 4.3.2

Multiply $4$ by $2$ .

$(5 m^{4} - n^{5}) (25 m^{8} + 5 m^{4} n^{5} + {(n^{5})}^{2})$

Step 4.4

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

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Step 4.4.1

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

$(5 m^{4} - n^{5}) (25 m^{8} + 5 m^{4} n^{5} + n^{5 \cdot 2})$

Step 4.4.2

Multiply $5$ by $2$ .

$(5 m^{4} - n^{5}) (25 m^{8} + 5 m^{4} n^{5} + n^{10})$