Pre-Algebra Examples

$125 m^{6} - 216 n^{6}$

Step 1

Rewrite $125 m^{6}$ as ${(5 m^{2})}^{3}$ .

${(5 m^{2})}^{3} - 216 n^{6}$

Step 2

Rewrite $216 n^{6}$ as ${(6 n^{2})}^{3}$ .

${(5 m^{2})}^{3} - {(6 n^{2})}^{3}$

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 $a = 5 m^{2}$ and $b = 6 n^{2}$ .

$(5 m^{2} - (6 n^{2})) ({(5 m^{2})}^{2} + 5 m^{2} (6 n^{2}) + {(6 n^{2})}^{2})$

Step 4

Simplify.

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Step 4.1

Multiply $6$ by $- 1$ .

$(5 m^{2} - 6 n^{2}) ({(5 m^{2})}^{2} + 5 m^{2} (6 n^{2}) + {(6 n^{2})}^{2})$

Step 4.2

Apply the product rule to $5 m^{2}$ .

$(5 m^{2} - 6 n^{2}) (5^{2} {(m^{2})}^{2} + 5 m^{2} (6 n^{2}) + {(6 n^{2})}^{2})$

Step 4.3

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

$(5 m^{2} - 6 n^{2}) (25 {(m^{2})}^{2} + 5 m^{2} (6 n^{2}) + {(6 n^{2})}^{2})$

Step 4.4

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

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Step 4.4.1

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

$(5 m^{2} - 6 n^{2}) (25 m^{2 \cdot 2} + 5 m^{2} (6 n^{2}) + {(6 n^{2})}^{2})$

Step 4.4.2

Multiply $2$ by $2$ .

$(5 m^{2} - 6 n^{2}) (25 m^{4} + 5 m^{2} (6 n^{2}) + {(6 n^{2})}^{2})$

Step 4.5

Rewrite using the commutative property of multiplication.

$(5 m^{2} - 6 n^{2}) (25 m^{4} + 5 \cdot 6 m^{2} n^{2} + {(6 n^{2})}^{2})$

Step 4.6

Multiply $5$ by $6$ .

$(5 m^{2} - 6 n^{2}) (25 m^{4} + 30 m^{2} n^{2} + {(6 n^{2})}^{2})$

Step 4.7

Apply the product rule to $6 n^{2}$ .

$(5 m^{2} - 6 n^{2}) (25 m^{4} + 30 m^{2} n^{2} + 6^{2} {(n^{2})}^{2})$

Step 4.8

Raise $6$ to the power of $2$ .

$(5 m^{2} - 6 n^{2}) (25 m^{4} + 30 m^{2} n^{2} + 36 {(n^{2})}^{2})$

Step 4.9

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

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Step 4.9.1

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

$(5 m^{2} - 6 n^{2}) (25 m^{4} + 30 m^{2} n^{2} + 36 n^{2 \cdot 2})$

Step 4.9.2

Multiply $2$ by $2$ .

$(5 m^{2} - 6 n^{2}) (25 m^{4} + 30 m^{2} n^{2} + 36 n^{4})$