Basic Math Examples

$\frac{{(s + t)}^{2}}{s - t} \cdot \frac{s^{3} - t^{3}}{s^{2} - t^{2}} \div \frac{s^{2} + s t + t^{2}}{{(s - t)}^{2}}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{{(s + t)}^{2}}{s - t} \cdot \frac{s^{3} - t^{3}}{s^{2} - t^{2}} \frac{{(s - t)}^{2}}{s^{2} + s t + t^{2}}$

Step 2

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 $a = s$ and $b = t$ .

$\frac{{(s + t)}^{2}}{s - t} \cdot \frac{(s - t) (s^{2} + s t + t^{2})}{s^{2} - t^{2}} \frac{{(s - t)}^{2}}{s^{2} + s t + t^{2}}$

Step 3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = s$ and $b = t$ .

$\frac{{(s + t)}^{2}}{s - t} \cdot \frac{(s - t) (s^{2} + s t + t^{2})}{(s + t) (s - t)} \frac{{(s - t)}^{2}}{s^{2} + s t + t^{2}}$

Step 4

Simplify terms.

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Step 4.1

Cancel the common factor of $s + t$ .

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Step 4.1.1

Factor $s + t$ out of ${(s + t)}^{2}$ .

$\frac{(s + t) (s + t)}{s - t} \cdot \frac{(s - t) (s^{2} + s t + t^{2})}{(s + t) (s - t)} \frac{{(s - t)}^{2}}{s^{2} + s t + t^{2}}$

Step 4.1.2

Cancel the common factor.

$\frac{(s + t) (s + t)}{s - t} \cdot \frac{(s - t) (s^{2} + s t + t^{2})}{(s + t) (s - t)} \frac{{(s - t)}^{2}}{s^{2} + s t + t^{2}}$

Step 4.1.3

Rewrite the expression.

$\frac{s + t}{s - t} \cdot \frac{(s - t) (s^{2} + s t + t^{2})}{s - t} \frac{{(s - t)}^{2}}{s^{2} + s t + t^{2}}$

Step 4.2

Cancel the common factor of $s - t$ .

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Step 4.2.1

Cancel the common factor.

$\frac{s + t}{s - t} \cdot \frac{(s - t) (s^{2} + s t + t^{2})}{s - t} \frac{{(s - t)}^{2}}{s^{2} + s t + t^{2}}$

Step 4.2.2

Rewrite the expression.

$(s + t) \cdot \frac{s^{2} + s t + t^{2}}{s - t} \frac{{(s - t)}^{2}}{s^{2} + s t + t^{2}}$

Step 4.3

Multiply $s + t$ by $\frac{s^{2} + s t + t^{2}}{s - t}$ .

$\frac{(s + t) (s^{2} + s t + t^{2})}{s - t} \cdot \frac{{(s - t)}^{2}}{s^{2} + s t + t^{2}}$

Step 4.4

Cancel the common factor of $s^{2} + s t + t^{2}$ .

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Step 4.4.1

Factor $s^{2} + s t + t^{2}$ out of $(s + t) (s^{2} + s t + t^{2})$ .

$\frac{(s^{2} + s t + t^{2}) (s + t)}{s - t} \cdot \frac{{(s - t)}^{2}}{s^{2} + s t + t^{2}}$

Step 4.4.2

Cancel the common factor.

$\frac{(s^{2} + s t + t^{2}) (s + t)}{s - t} \cdot \frac{{(s - t)}^{2}}{s^{2} + s t + t^{2}}$

Step 4.4.3

Rewrite the expression.

$\frac{s + t}{s - t} {(s - t)}^{2}$

Step 4.5

Cancel the common factor of $s - t$ .

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Step 4.5.1

Factor $s - t$ out of ${(s - t)}^{2}$ .

$\frac{s + t}{s - t} ((s - t) (s - t))$

Step 4.5.2

Cancel the common factor.

$\frac{s + t}{s - t} ((s - t) (s - t))$

Step 4.5.3

Rewrite the expression.

$(s + t) (s - t)$

Step 5

Expand $(s + t) (s - t)$ using the FOIL Method.

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Step 5.1

Apply the distributive property.

$s (s - t) + t (s - t)$

Step 5.2

Apply the distributive property.

$s \cdot s + s (- t) + t (s - t)$

Step 5.3

Apply the distributive property.

$s \cdot s + s (- t) + t s + t (- t)$

Step 6

Simplify terms.

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Step 6.1

Combine the opposite terms in $s \cdot s + s (- t) + t s + t (- t)$ .

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Step 6.1.1

Reorder the factors in the terms $s (- t)$ and $t s$ .

$s \cdot s - s t + s t + t (- t)$

Step 6.1.2

Add $- s t$ and $s t$ .

$s \cdot s + 0 + t (- t)$

Step 6.1.3

Add $s \cdot s$ and $0$ .

$s \cdot s + t (- t)$

Step 6.2

Simplify each term.

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Step 6.2.1

Multiply $s$ by $s$ .

$s^{2} + t (- t)$

Step 6.2.2

Rewrite using the commutative property of multiplication.

$s^{2} - t \cdot t$

Step 6.2.3

Multiply $t$ by $t$ by adding the exponents.

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Step 6.2.3.1

Move $t$ .

$s^{2} - (t \cdot t)$

Step 6.2.3.2

Multiply $t$ by $t$ .

$s^{2} - t^{2}$