Basic Math Examples

$\frac{{(v + 9)}^{3}}{{(v + 7)}^{3}} \cdot \frac{v^{2} + 14 v + 49}{v^{2} + 18 v + 81}$

Step 1

Factor using the perfect square rule.

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Step 1.1

Rewrite $49$ as $7^{2}$ .

$\frac{{(v + 9)}^{3}}{{(v + 7)}^{3}} \cdot \frac{v^{2} + 14 v + 7^{2}}{v^{2} + 18 v + 81}$

Step 1.2

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

$14 v = 2 \cdot v \cdot 7$

Step 1.3

Rewrite the polynomial.

$\frac{{(v + 9)}^{3}}{{(v + 7)}^{3}} \cdot \frac{v^{2} + 2 \cdot v \cdot 7 + 7^{2}}{v^{2} + 18 v + 81}$

Step 1.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = v$ and $b = 7$ .

$\frac{{(v + 9)}^{3}}{{(v + 7)}^{3}} \cdot \frac{{(v + 7)}^{2}}{v^{2} + 18 v + 81}$

Step 2

Factor using the perfect square rule.

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Step 2.1

Rewrite $81$ as $9^{2}$ .

$\frac{{(v + 9)}^{3}}{{(v + 7)}^{3}} \cdot \frac{{(v + 7)}^{2}}{v^{2} + 18 v + 9^{2}}$

Step 2.2

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

$18 v = 2 \cdot v \cdot 9$

Step 2.3

Rewrite the polynomial.

$\frac{{(v + 9)}^{3}}{{(v + 7)}^{3}} \cdot \frac{{(v + 7)}^{2}}{v^{2} + 2 \cdot v \cdot 9 + 9^{2}}$

Step 2.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = v$ and $b = 9$ .

$\frac{{(v + 9)}^{3}}{{(v + 7)}^{3}} \cdot \frac{{(v + 7)}^{2}}{{(v + 9)}^{2}}$

Step 3

Simplify terms.

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Step 3.1

Combine.

$\frac{{(v + 9)}^{3} {(v + 7)}^{2}}{{(v + 7)}^{3} {(v + 9)}^{2}}$

Step 3.2

Cancel the common factor of ${(v + 9)}^{3}$ and ${(v + 9)}^{2}$ .

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Step 3.2.1

Factor ${(v + 9)}^{2}$ out of ${(v + 9)}^{3} {(v + 7)}^{2}$ .

$\frac{{(v + 9)}^{2} ((v + 9) {(v + 7)}^{2})}{{(v + 7)}^{3} {(v + 9)}^{2}}$

Step 3.2.2

Cancel the common factors.

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Step 3.2.2.1

Factor ${(v + 9)}^{2}$ out of ${(v + 7)}^{3} {(v + 9)}^{2}$ .

$\frac{{(v + 9)}^{2} ((v + 9) {(v + 7)}^{2})}{{(v + 9)}^{2} {(v + 7)}^{3}}$

Step 3.2.2.2

Cancel the common factor.

$\frac{{(v + 9)}^{2} ((v + 9) {(v + 7)}^{2})}{{(v + 9)}^{2} {(v + 7)}^{3}}$

Step 3.2.2.3

Rewrite the expression.

$\frac{(v + 9) {(v + 7)}^{2}}{{(v + 7)}^{3}}$

Step 3.3

Cancel the common factor of ${(v + 7)}^{2}$ and ${(v + 7)}^{3}$ .

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Step 3.3.1

Factor ${(v + 7)}^{2}$ out of $(v + 9) {(v + 7)}^{2}$ .

$\frac{{(v + 7)}^{2} (v + 9)}{{(v + 7)}^{3}}$

Step 3.3.2

Cancel the common factors.

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Step 3.3.2.1

Factor ${(v + 7)}^{2}$ out of ${(v + 7)}^{3}$ .

$\frac{{(v + 7)}^{2} (v + 9)}{{(v + 7)}^{2} (v + 7)}$

Step 3.3.2.2

Cancel the common factor.

$\frac{{(v + 7)}^{2} (v + 9)}{{(v + 7)}^{2} (v + 7)}$

Step 3.3.2.3

Rewrite the expression.

$\frac{v + 9}{v + 7}$