Finite Math Examples

$64 m^{6} - 125 n^{6}$

Step 1

Rewrite $64 m^{6}$ as ${(4 m^{2})}^{3}$ .

${(4 m^{2})}^{3} - 125 n^{6}$

Step 2

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

${(4 m^{2})}^{3} - {(5 n^{2})}^{3}$

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 $a = 4 m^{2}$ and $b = 5 n^{2}$ .

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

Step 4

Simplify.

Tap for more steps...

Step 4.1

Multiply $5$ by $- 1$ .

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

Step 4.2

Apply the product rule to $4 m^{2}$ .

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

Step 4.3

Raise $4$ to the power of $2$ .

$(4 m^{2} - 5 n^{2}) (16 {(m^{2})}^{2} + 4 m^{2} (5 n^{2}) + {(5 n^{2})}^{2})$

Step 4.4

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

Tap for more steps...

Step 4.4.1

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

$(4 m^{2} - 5 n^{2}) (16 m^{2 \cdot 2} + 4 m^{2} (5 n^{2}) + {(5 n^{2})}^{2})$

Step 4.4.2

Multiply $2$ by $2$ .

$(4 m^{2} - 5 n^{2}) (16 m^{4} + 4 m^{2} (5 n^{2}) + {(5 n^{2})}^{2})$

Step 4.5

Rewrite using the commutative property of multiplication.

$(4 m^{2} - 5 n^{2}) (16 m^{4} + 4 \cdot 5 m^{2} n^{2} + {(5 n^{2})}^{2})$

Step 4.6

Multiply $4$ by $5$ .

$(4 m^{2} - 5 n^{2}) (16 m^{4} + 20 m^{2} n^{2} + {(5 n^{2})}^{2})$

Step 4.7

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

$(4 m^{2} - 5 n^{2}) (16 m^{4} + 20 m^{2} n^{2} + 5^{2} {(n^{2})}^{2})$

Step 4.8

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

$(4 m^{2} - 5 n^{2}) (16 m^{4} + 20 m^{2} n^{2} + 25 {(n^{2})}^{2})$

Step 4.9

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

Tap for more steps...

Step 4.9.1

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

$(4 m^{2} - 5 n^{2}) (16 m^{4} + 20 m^{2} n^{2} + 25 n^{2 \cdot 2})$

Step 4.9.2

Multiply $2$ by $2$ .

$(4 m^{2} - 5 n^{2}) (16 m^{4} + 20 m^{2} n^{2} + 25 n^{4})$