**6.2.2 Algebraic Expressions (III), PT3 Practice**

Question 6:

Question 6:

(a) Simplify each of the following:

$\begin{array}{l}\text{(i)}\frac{12mn}{32}\\ \text{(ii)}\frac{{x}^{2}-xy}{x}\end{array}$

(b) Express $\frac{1}{2q}-\frac{2p-7}{6q}$ as a single fraction in its simplest form.

(b) Express $\frac{1}{2q}-\frac{2p-7}{6q}$ as a single fraction in its simplest form.

Solution:Solution:

$\begin{array}{l}\text{(a)(i)}\frac{12mn}{32}=\frac{3mn}{8}\\ \text{(a)(ii)}\frac{{x}^{2}-xy}{x}=\frac{\overline{)x}\left(x-y\right)}{\overline{)x}}=x-y\end{array}$

(b)

(b)

$\begin{array}{l}\frac{1}{2q}-\frac{2p-7}{6q}=\frac{1\times 3}{2q\times 3}-\frac{\left(2p-7\right)}{6q}\\ \text{}=\frac{3-2p+7}{6q}\\ \text{}=\frac{10-2p}{6q}\\ \text{}=\frac{\overline{)2}\left(5-p\right)}{3\overline{)6}q}\\ \text{}=\frac{5-p}{3q}\end{array}$

**Question 7:**

*ae*+ 3

*af*– 6

*de*– 9

*df*(a) Factorise 2

$\frac{{a}^{2}-{b}^{2}}{{(a+b)}^{2}}$
(b) Simplify

Solution:Solution:

**(a)**

2

*ae*+ 3*af*– 6*de*– 9*df*=*a*(2*e*+ 3*f*)*– 3**d*(2*e*+ 3*f*)**= (2**

*e*+ 3*f*)*(**a*– 3*d*)

(b)

(b)

$\begin{array}{l}\frac{{a}^{2}-{b}^{2}}{{(a+b)}^{2}}=\frac{\overline{)(a+b)}(a-b)}{\overline{)(a+b)}(a+b)}\\ \text{}=\frac{a-b}{a+b}\end{array}$

**Question 8:**

*c*

^{2}– 12

*ac.*(a) Factorise –8

$\frac{ae+ad-2be-2bd}{{a}^{2}-4{b}^{2}}.$
(b) Simplify

Solution:Solution:

**(a)**

–8

*c*^{2}– 12*ac**=*

**–4**

*c*(2*c*+ 3*a*)

(b)

(b)

$\begin{array}{l}\frac{ae+ad-2be-2bd}{{a}^{2}-4{b}^{2}}=\frac{a\left(e+d\right)-2b\left(e+d\right)}{\left(a+2b\right)\left(a-2b\right)}\\ \text{}=\frac{\left(e+d\right)\overline{)\left(a-2b\right)}}{\left(a+2b\right)\overline{)\left(a-2b\right)}}\\ \text{}=\frac{e+d}{a+2b}\end{array}$

**Question 9:**

*x*

^{2}– 27

*y*

^{2}(a) Factorise 12

$\frac{3{m}^{2}-10m+3}{{m}^{2}-9}\xf7\frac{3m-1}{m+3}.$
(b) Simplify

Solution:Solution:

**(a)**

12

*x*^{2 }– 27*y*^{2 }= 3 (4*x*^{2}– 9*y*^{2})**= 3(2**

*x*– 3*y*) (2*x*+ 3*y*)

(b)

(b)

$\begin{array}{l}\frac{3{m}^{2}-10m+3}{{m}^{2}-9}\xf7\frac{3m-1}{m+3}=\frac{\overline{)\left(3m-1\right)}\overline{)\left(m-3\right)}}{\overline{)\left(m+3\right)}\overline{)\left(m-3\right)}}\times \frac{\overline{)m+3}}{\overline{)3m-1}}\\ \text{}=1\end{array}$

**Question 10:**

$\text{Simplify:}\frac{8m+mn}{3m}\xf7\frac{{n}^{2}-64}{24}$

*Solution:*$\begin{array}{l}\frac{8m+mn}{3m}\xf7\frac{{n}^{2}-64}{24}\\ =\frac{8m+mn}{3m}\times \frac{24}{{n}^{2}-64}\\ =\frac{m\left(8+n\right)}{3m}\times \frac{24}{{n}^{2}-{8}^{2}}\\ =\frac{\overline{)m}\overline{)\left(8+n\right)}}{\overline{)3}\overline{)m}}\times \frac{{\overline{)24}}_{8}}{\left(n-8\right)\overline{)\left(n+8\right)}}\\ =\frac{8}{n-8}\end{array}$