Basic Math Examples

$\frac{t^{2} + 2}{t^{2} + 3 t} \div \frac{t^{2}}{t^{2} - 6 t + 9}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{t^{2} + 2}{t^{2} + 3 t} \cdot \frac{t^{2} - 6 t + 9}{t^{2}}$

Step 2

Factor $t$ out of $t^{2} + 3 t$ .

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Step 2.1

Factor $t$ out of $t^{2}$ .

$\frac{t^{2} + 2}{t \cdot t + 3 t} \cdot \frac{t^{2} - 6 t + 9}{t^{2}}$

Step 2.2

Factor $t$ out of $3 t$ .

$\frac{t^{2} + 2}{t \cdot t + t \cdot 3} \cdot \frac{t^{2} - 6 t + 9}{t^{2}}$

Step 2.3

Factor $t$ out of $t \cdot t + t \cdot 3$ .

$\frac{t^{2} + 2}{t (t + 3)} \cdot \frac{t^{2} - 6 t + 9}{t^{2}}$

Step 3

Factor using the perfect square rule.

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Step 3.1

Rewrite $9$ as $3^{2}$ .

$\frac{t^{2} + 2}{t (t + 3)} \cdot \frac{t^{2} - 6 t + 3^{2}}{t^{2}}$

Step 3.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$6 t = 2 \cdot t \cdot 3$

Step 3.3

Rewrite the polynomial.

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

Step 3.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = t$ and $b = 3$ .

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

Step 4

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

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Step 4.1

Multiply $\frac{t^{2} + 2}{t (t + 3)}$ by $\frac{{(t - 3)}^{2}}{t^{2}}$ .

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

Step 4.2

Multiply $t$ by $t^{2}$ by adding the exponents.

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Step 4.2.1

Move $t^{2}$ .

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

Step 4.2.2

Multiply $t^{2}$ by $t$ .

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Step 4.2.2.1

Raise $t$ to the power of $1$ .

$\frac{(t^{2} + 2) {(t - 3)}^{2}}{t^{2} t^{1} (t + 3)}$

Step 4.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{(t^{2} + 2) {(t - 3)}^{2}}{t^{2 + 1} (t + 3)}$

Step 4.2.3

Add $2$ and $1$ .

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