The algebra of the four basic operations  (Page 5/5)

$\frac{{\mathrm{4x}}^{3}y}{{\mathrm{6y}}^2}÷\frac{\text{xy}}{{\mathrm{3x}}^2}\times \frac{2{\text{xy}}^2}{\mathrm{3x}}$ = $\frac{{\mathrm{4x}}^{3}y}{{\mathrm{6y}}^2}\times \frac{{\mathrm{3x}}^2}{\text{xy}}\times \frac{2{\text{xy}}^2}{\mathrm{3x}}$ = $\frac{{\mathrm{4x}}^4}{3}$

$\frac{a^2-9}{2}\times \frac{1}{{\mathrm{4a}}^2-\text{12}a}$ = $\frac{\left(a+3\right)\left(a-3\right)}{2}\times \frac{1}{\mathrm{4a}\left(a-3\right)}$ = $\frac{\left(a+3\right)}{\mathrm{8a}}$

$\frac{\mathrm{3a}+6}{5}÷\frac{a^2-4}{\text{10}}$ = $\frac{\mathrm{3a}+6}{5}\times \frac{\text{10}}{a^2-4}$ = $\frac{3\left(a+2\right)}{5}\times \frac{\text{10}}{\left(a+2\right)\left(a-2\right)}$ = $\frac{6}{a-2}$

Exercise:

Simplify:

1. $\frac{2{\text{ab}}^2}{b^{3}c}\times \frac{9{\text{ac}}^2}{\mathrm{4b}}÷\frac{3\text{ac}}{{\mathrm{2b}}^2}$

2. $\frac{2\left(a+1\right){\left(a-2\right)}^2}{{\left(a-2\right)}^3\left(a+3\right)}\times \frac{9\left(a+1\right){\left(a+3\right)}^2}{4\left(a-2\right)}÷\frac{3\left(a+1\right)\left(a+3\right)}{2{\left(a-2\right)}^2}$

3. $\frac{{\mathrm{4a}}^2+\mathrm{8a}}{\mathrm{2b}+4}\times \frac{3\left(b^2+2\right)}{{\mathrm{3a}}^2+\mathrm{6a}}$

4. $\frac{x^2-1}{\mathrm{5x}-5}÷\frac{{\left(x+1\right)}^2}{\text{15}x+\text{15}}$

5. $\frac{\mathrm{7x}}{3\text{xy}}÷\frac{\mathrm{3x}+6}{{\mathrm{5x}}^{2}y}÷\frac{\mathrm{5x}-\text{10}}{{\mathrm{3x}}^2-\text{12}}$

6. $\frac{\frac{{\mathrm{5x}}^2+\mathrm{5x}}{x^2-x}}{\frac{\mathrm{5x}+5}{x^2-1}}$ (here we have a fraction divided by a fraction – first rewrite it like number 4)

C. Adding fractions

You already know quite well that adding and subtracting fractions is a lot more difficult than multiplying and dividing them. The reason is that we can only add and subtract the same kind of fractions, namely fractions with identical denominators. If the denominators are different, find the lowest common multiple of the denominators (LCD), rewrite all the terms with this as denominator, and then simplify by gathering like terms together. Finally, the answer has to be simplified by cancelling factors occurring in both numerator and denominator. Here are some examples – all the steps have been shown:

Simplify:

1. $\frac{5\text{abx}}{2\text{cx}}+\frac{4\text{ac}}{\mathrm{3x}}+\frac{\text{cx}}{\mathrm{2a}}$ (LCD = 6acx)

$\left(\frac{5\text{abx}}{2\text{cx}}\times \frac{\mathrm{3a}}{\mathrm{3a}}\right)+\left(\frac{4\text{ac}}{\mathrm{3x}}\times \frac{2\text{ac}}{2\text{ac}}\right)+\left(\frac{\text{cx}}{\mathrm{2a}}\times \frac{3\text{cx}}{3\text{cx}}\right)$ = $\frac{\text{15}{a}^2\text{bx}}{6\text{acx}}+\frac{{\mathrm{8a}}^2{c}^2}{6\text{acx}}+\frac{{\mathrm{3c}}^2{x}^2}{6\text{acx}}$ = $\frac{\text{15}{a}^2\text{bx}+{\mathrm{8a}}^2{c}^2+{\mathrm{3c}}^2{x}^2}{6\text{acx}}$

2. $\frac{a+b}{2}+\frac{b+c}{3}-\frac{a+c}{6}$ (LCD = 6) Take careful note of how the signs are handled below!

$\frac{3\left(a+b\right)}{6}+\frac{2\left(b+c\right)}{6}-\frac{a+c}{6}$ = $\frac{3\left(a+b\right)+2\left(b+c\right)-\left(a+c\right)}{6}$ = $\frac{\mathrm{3a}+\mathrm{3b}+\mathrm{2b}+\mathrm{2c}-a-c}{6}$ = $\frac{\mathrm{2a}+\mathrm{5b}+c}{6}$

3. $\frac{a+3}{a^2-4}+\frac{1}{\mathrm{3a}+6}+\frac{2}{\mathrm{5a}-\text{10}}$

To find Lowest Common Denominator first factorise denominators!

$\frac{a+3}{\left(a+2\right)\left(a-2\right)}+\frac{1}{3\left(a+2\right)}+\frac{2}{5\left(a-2\right)}$

Do you see that the LCD is 3×5×(a+2)(a–2)?

$\left(\frac{a+3}{\left(a+2\right)\left(a-2\right)}\times \frac{\text{15}}{\text{15}}\right)+\left(\frac{1}{3\left(a+2\right)}\times \frac{5\left(a-2\right)}{5\left(a-2\right)}\right)+\left(\frac{2}{5\left(a-2\right)}\times \frac{3\left(a+2\right)}{3\left(a+2\right)}\right)$

= $\frac{\text{15}\left(a+3\right)+5\left(a-2\right)+6\left(a+2\right)}{\text{15}\left(a+2\right)\left(a-2\right)}$ = $\frac{\text{15}a+\text{45}+\mathrm{5a}-\text{10}+\mathrm{6a}+\text{12}}{\text{15}\left(a+2\right)\left(a-2\right)}$ = $\frac{\text{26}a+\text{47}}{\text{15}\left(a+2\right)\left(a-2\right)}$

Exercise:

Simplify the following expressions by using factorising:

1. $\frac{a}{x^2}-\frac{a}{x}+\frac{\mathrm{5a}}{\mathrm{2x}}$

2. $\frac{1}{3}+\frac{\mathrm{2x}+1}{\mathrm{2x}}-\frac{x-1}{\mathrm{3x}}$

3. $\frac{\mathrm{4a}-\mathrm{4b}}{{\mathrm{2a}}^2-{\mathrm{2b}}^2}-\frac{3}{\mathrm{2a}-\mathrm{2b}}$

4. $\frac{1}{2}\left(a-2\right)+\frac{2}{3}\left(a+1\right)-\frac{3}{4}\left(a-3\right)$

Assessment: Assess the 4 problems in the exercise above.

Here is one final trick. We could simplify $\frac{2\left(x-1\right)}{3\left(x+3\right)}\times \frac{9\left(x+3\right)}{\left(1-x\right)}$ better if (1–x) had been: (x–1).

So, we make the change we want by changing the sign of the whole binomial as well:

(1–x) = –(x–1) because –(x–1) = –x + 1, which is 1–x. Finish the problem yourself.

Assessment

Memorandum

TEST 1

1. Simplify the following expressions by collecting like terms:

1.1 3a2 + 3a2 – 6a + 3a – 4 + 1

1.2 2y2 – 1y + 2y2 – 6 + 2y – 9

1.3 8x2 – (5x + 12x2 – 1) + x – 4

1.4 (3a – a2) – [(2a2 – 11) – (5a – 3)]

2. Give the answers to the following problems in the simplest form:

2.1 Add 3x2 + 5x – 1 to x2 – 3x

2.2 Find the sum of 2a + 3b – 5 and 3 + 2b – 7a

2.3 Subtract 6a + 7 from 5a2 + 2a + 2

2.4 How much is 3a – 8b + 3 less than a + b + 2?

3. Simplify by multiplication, leaving your answer in the simplest form:

3.1 (3x2) × (2x3)

3.2 (abc) (a2c) (2b2)

3.3 abc(a2c + 2b2)

3.4 –3a(2a2 – 5a)

3.5 (a – 2b) (a + 2b)

3.6 (3 – x2) (2x2 + 5)

3.7 (x – 5y)2

3.8 (2 – b) (3a + c)

TEST 1 – Memorandum

1.1 6a2 – 3a – 3

1.2 4y2 + y – 15

1.3 – 4x2 – 4x – 3

1.4 – 3a2 + 8a + 8

2.1 4x2 + 2x – 1

2.2 – 5a + 5b – 2

2.3 5a2 – 4a – 5

2.4 – 2a + 9b – 1

3.1 6x5

3.2 2a3b3c2

3.3 a3bc2 + 2ab3c

3.4 – 6a3 + 15a2

3.5 a2 – 4b2

3.6 –2x4 + x2 + 15

3.7 x2 – 10xy + 25y2

3.8 6a + 2c – 3ab – bc

TEST 2

1. Find the Highest Common Factor of these three expressions: 6a2c2 and 2ac2 and 10ab2c3.

2. Completely factorise these expressions by finding common factors:

2.1 12a3 + 3a4

2.2 –5xy – 15x2y2 – 20y

2.3 6a2c2 – 2ac2 + 10ab2c3

3. Factorise these differences of squares completely:

3.1 a2 – 4

3.2 $\frac{1}{9}{a}^2-{\mathrm{9b}}^2$

3.3 x4 – 16y4

3.4 1 – a4b4

4. Factorise these expressions as far as possible:

4.1 3x2 – 27

4.2 2a – 8ab2

4.3 a2 – 5a – 6

4.4 a2 + 7a + 6

5. Simplify the following fractions by making use of factorising:

5.1 $\frac{{\mathrm{3a}}^2-3}{\mathrm{6a}+6}$

5.2 $\frac{{\mathrm{6x}}^{2}y-\mathrm{6y}}{\mathrm{2x}-2}$

5.3 $\frac{a^2-9}{2}\times \frac{1}{{\mathrm{4a}}^2-\text{12}a}$

5.4 $\frac{\mathrm{3x}+6}{5}÷\frac{x^2-4}{\text{15}}$

5.5 $\frac{\text{abx}}{2\text{cx}}+\frac{2\text{ac}}{\mathrm{3x}}+\frac{3\text{cx}}{\mathrm{2a}}$

5.6 $\frac{\mathrm{2a}}{x^2}-\frac{\mathrm{3a}}{x}+\frac{a}{\mathrm{2x}}$

5.7 $\frac{\mathrm{4a}-\mathrm{4b}}{{\mathrm{2a}}^2-{\mathrm{2b}}^2}-\frac{3}{\mathrm{2a}-\mathrm{2b}}$

5.8 $\frac{2}{3}\left(a+2\right)+\frac{1}{3}\left(a-1\right)-\frac{1}{4}\left(a-5\right)$

TEST 2 – Memorandum

1. 2ac2

2.1 3a3 (4 + a2)

2.2 –5y (x + 3x2y + 4)

2.3 2ac2 (3a – 1 + 5b2c)

3.1 (a + 2) (a – 2)

3.2 $\left(\frac{1}{3}a+\mathrm{3b}\right)\left(\frac{1}{3}a-\mathrm{3b}\right)$

3.3 (x2 + 4y2) (x + 2y) (x – 2y)

3.4 (1 + a2b2) (1 + ab) (1 – ab)

4.1 3 (x + 3) (x – 3)

4.2 2a (1 + 4b) (1 – 4b)

4.3 (a + 1) (a – 6)

4.4 (a + 1) (a + 6)

5.1 $\frac{a-1}{2}$

5.2 3y (x + 1)

5.3 $\frac{a+3}{\mathrm{8a}}$

5.4 $\frac{9}{x-2}$

5.5 $\frac{{\mathrm{3a}}^2\text{bx}+{\mathrm{4a}}^2{c}^2+{\mathrm{9c}}^2{x}^2}{6\text{acx}}$

5.6 $\frac{\mathrm{4a}-5\text{ax}}{{\mathrm{2x}}^2}$

5.7 $\frac{a-\mathrm{7b}}{2\left(a+b\right)\left(a-b\right)}$

5.8 $\frac{\mathrm{3a}+9}{4}$