# Simplify the following: " "a: color(white)("d")(x^2+7x+12)/(x+3) " "b: color(white)("d") sqrt((x^9y^4)/x^5) " "c: (3^(-1)a^4b^(-3))^2/(6a^2b^(-1)c^(-2))^2 ?

Apr 4, 2017

Solution to a) $\to x + 4$

#### Explanation:

Solution to a)

Factorising takes quite a bit of practice as you need to build up a 'memory base' that you can draw on.

Given:$\text{ } \frac{{x}^{2} + 7 x + 12}{x + 3}$

Consider the top bit (numerator)

Notice that $3 \times 4 = 12 \text{ and that } 3 + 4 = 7$

The coefficient of ${x}^{2}$ (number in front of it) is 1
so we can write ${x}^{2} + 7 x + 12 \text{ as } \left(x + 3\right) \left(x + 4\right)$

Putting it all together we have:

$\frac{{x}^{2} + 7 x + 12}{x + 3} \text{ "=" } \frac{\left(x + 3\right) \left(x + 4\right)}{\left(x + 3\right)}$

$\text{ "=" } \frac{x + 3}{x + 3} \times \left(x + 4\right)$

$\text{ "=" "1" } \times \left(x + 4\right)$

$\text{ "=" } x + 4$

Apr 4, 2017

Solution to b) $\to {x}^{2} {y}^{2}$

#### Explanation:

Given:" "sqrt((x^9y^4)/x^5

You are looking for squared values as these can be 'taken outside' the root. Also note that (by example) $\sqrt{\frac{a}{b}} \to \frac{\sqrt{a}}{\sqrt{b}}$

Write as: (sqrt(cancel(x^2)xx cancel(x^2)xx x^2xx x^2xx x xxy^2xxy^2))/(sqrt(cancel(x^2)xx cancel(x^2)xx x)

$= \frac{{x}^{2} {y}^{2} \cancel{\sqrt{x}}}{\cancel{\sqrt{x}}}$

$= {x}^{2} {y}^{2}$

Apr 4, 2017

Solution to c) $\text{ } \frac{{b}^{8} {c}^{4}}{4 {a}^{16}}$

#### Explanation:

By example:
Note that ${a}^{- 1} = \frac{1}{a} \text{; "a^(-2)=1/a^2"; } {a}^{- \frac{2}{3}} = \frac{1}{\sqrt[3]{{a}^{2}}}$

Note that $\frac{1}{{a}^{- 1}} = a \text{; "1/a^(-2)=a^2"; } \frac{1}{a} ^ \left(- \frac{2}{3}\right) = \sqrt[3]{{a}^{2}}$
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$\textcolor{b l u e}{\text{Consider the numerator:}}$
$\text{ "(3^(-1)a^4b^(-3))^(-2)" "->" } {\left(\frac{1}{3} \times {a}^{4} \times \frac{1}{b} ^ 3\right)}^{- 2}$

$\text{ " = " } {\left({a}^{4} / \left(3 {b}^{3}\right)\right)}^{- 2}$

$\text{ " = " } {\left(\frac{3 {b}^{3}}{a} ^ 4\right)}^{2}$

$\text{ " = " } \frac{9 {b}^{6}}{a} ^ \left(12\right)$
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$\textcolor{b l u e}{\text{Consider the denominator:}}$
$\text{ } {\left(6 {a}^{2} {b}^{- 1} {c}^{- 2}\right)}^{2} \to {\left(6 \times {a}^{2} \times \frac{1}{b} \times \frac{1}{c} ^ 2\right)}^{2}$

$\text{ "=" } {\left(\frac{6 {a}^{2}}{b {c}^{2}}\right)}^{2}$

$\text{ "=" } \frac{36 {a}^{4}}{{b}^{2} {c}^{4}}$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Putting it all together}}$

$\textcolor{b r o w n}{\text{numerator " -:" denominator " ->" } \frac{9 {b}^{6}}{{a}^{12}} \div \frac{36 {a}^{4}}{{b}^{2} {c}^{4}}}$

$\text{ } \frac{9 {b}^{6}}{{a}^{12}} \times \frac{{b}^{2} {c}^{4}}{36 {a}^{4}}$

$\text{ "=" } \frac{9}{36} \times \frac{{b}^{8} {c}^{4}}{a} ^ 16$

$\text{ "=" } \frac{{b}^{8} {c}^{4}}{4 {a}^{16}}$