Basic Math Examples

$y^{3 m} - 1$

Step 1

Rewrite $y^{3 m}$ as ${(y^{m})}^{3}$ .

${(y^{m})}^{3} - 1$

Step 2

Rewrite $1$ as $1^{3}$ .

${(y^{m})}^{3} - 1^{3}$

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 $a = y^{m}$ and $b = 1$ .

$(y^{m} - 1) ({(y^{m})}^{2} + y^{m} \cdot 1 + 1^{2})$

Step 4

Simplify.

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Step 4.1

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

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Step 4.1.1

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

$(y^{m} - 1) (y^{m \cdot 2} + y^{m} \cdot 1 + 1^{2})$

Step 4.1.2

Move $2$ to the left of $m$ .

$(y^{m} - 1) (y^{2 m} + y^{m} \cdot 1 + 1^{2})$

Step 4.2

Multiply $y^{m}$ by $1$ .

$(y^{m} - 1) (y^{2 m} + y^{m} + 1^{2})$

Step 4.3

One to any power is one.

$(y^{m} - 1) (y^{2 m} + y^{m} + 1)$