Algebraic Expressions

Exercise 9.2

Question 1: Find the product of the following pairs of monomials.

(i) 4, 7p

Answer: `4 xx 7 p = 28p`

(ii) – 4p, 7p

Answer: `- 4p xx 7p = -28p^2`

(iii) – 4p, 7pq

Answer: `- 4p xx 7pq = -28p^2q`

(iv) `4p^3`, – 3p

Answer: `4p^3q xx - 3p = -12p^4q`

(v) 4p, 0

Answer: `4p xx 0 = 0`

Question 2: Find the areas of rectangles with the following pairs of monomials as their lengths and breadths respectively.

`(p, q)`, `(10m, 5n)`, (`20x^2\5y^2`, `(4x, 3x^2`), (`3mn`, `4np`)

Answer: Area = Length × breadth

(i) `p xx q = pq`

(ii) `10m xx 5n = 50mn`

(iii) `20x^2 xx 5y^2 = 100x^2y^2`

(iv) `4x \xx 3x^2 = 12x^3`

(v) `3mn xx 4np = 12mn^2p`

Question 3: Complete the following table of products:

Answer:

Mononomials	2x	-5y	3x²	-4xy	7x²y	9x²y²
2x	4x²	-10xy	6x³	-8x²y	14x³y	-18x³y²
-5y	-10xy	25y²	-15x²y	20xy²	-35x²²	45x²y³
3x²	6x³	-15x²y	9x⁴	-12x³y	21x⁴y	-27x⁴y²
-4xy	-8x²y	20xy²	-12x³y	16x²y²	-28x³y²	36x³y³
7x²y	14x³y	-35x²y²	21x⁴y	-28x³y²	49x⁴y²	-63x⁴y³
-9x²y²	-18x³y²	45x²y³	-27x⁴y²	36x³y³	-63x⁴y³	81x⁴y⁴

Question 4: Obtain the volume of rectangular boxes with the following length, breadth and height respectively.

(i) `5a`, `3a^2`, `7a^4` (ii) `2p`, `4q`, `8r` (iii) `xy`, `2x^2y`, `2xy^2`(iv) `a`, `2b`, `3c`

Answer: Volume = length × breadth × height
(i) `5a \xx 3a^2 xx 7a^4 = 105a^7`
(ii) `2p \xx 4q \xx 8r = 64pqr`
(iii) `xy \xx 2x^2y \xx 2xy^2 = 4x^4y^4`
(iv) `a \xx 2b \xx 3c = 6abc`

Question 5: Obtain the product of

(i) xy, yz, zx (ii) a, – a², a³(iii) 2, 4y, 8y² 16y³(iv) a, 2b, 3c, 6abc (v) m, – mn, mnp

Answer: (i) `x^2y^2z^2`
(ii) `–a^6`
(iii) `1024y^6`
(iv) `36a^2b^2c^2`
(v) `–m^3n^2p`

Exercise 9.3

Question 1: Carry out the multiplication of the expressions in each of the following pairs.

(i) `4p`, `q + r`

Answer: `4p(q + r) = 4pq + 4pr`

(ii) `ab, `a – b`

Answer: `ab(a - b) = a^2b - ab^2`

(iii) `a + b`, `7a^2b^2`

Answer: `(a + b) (7a^2b^2) = 7a^3b^2 + 7a^2b^3`

(iv) `a^2– 9, `4a`

Answer: `(a^2 - 9)(4a) = 4a^3 - 36a^2`

(v) `pq + qr + rp`, 0

Answer: `(pq + qr + rp) xx 0 = 0`

Question 2: Complete the table.

Answer:

First expression	Second expression	Product
a	b + c + d	ab + ac + ad
x + y - 5	5xy	5x²y + 5xy² - 25xy
p	6p² - 7p + 5	6p^{3 - 7p^{2 + 5p}}
4p²q²	p² - q2	4p⁴q² - 4p²1⁴
a + b + c	abc	a²bc + ab²c + abc²

Question 3: Find the product.

(i) `a^2 xx (2a^22 xx (4a^26)`

Answer: As you know; `a^m \xx a^n \xx a^o = a^(m+n+o)`

So, we get; `a^2 xx 2a^22 xx 4a^26)= 8a^48`

(ii) `2/3xy\xx(-(9)/(10)x^2y^2)`

Answer: `=-3/5x^3y^3`

(iii) `(10)/(3)pq^3\xx6/5p^3q`

Answer: `=-4p^4q^4`

(iv) `x \xx\ x^2 xx\ x^3 xx \x^8`

Answer: `= x^14`

Question 4: (a) Simplify 3x (4x – 5) + 3 and find its values for (i) `x = 3` (ii) `x =1/2`

Answer:(i) putting `x = 3` in the equation we get
`12x^2 - 15x + 3`
`= 108 - 45 + 3 = 66`

(ii) putting `x = 1/2` in the equation we get

`12xx1/4-(15)/(2)+3=3-(15)/(2)+3=(15)/(2)`

Question 4: (b) Simplify `a (a^2+ a + 1) + 5` and find its value for (i) `a = 0`, (ii) `a = 1` and (iii) `a = – 1`

Answer: `a(a^2+a+1)`
`=a^3+a^2+a`

(i) putting `a= 0` in the equation we get
`0^3 + 0^2 + 0 = 0`

(ii) putting `a = 1` in the equation we get
`1^3+ 1^2+ 1 = 1 + 1 + 1 = 3`

(iii) putting `a = -1` in the equation we get
`-1^3 + 1^2 -1 = -1 + 1 + 1 = 1`

Question 5: (a) Add: `p ( p – q)`, `q ( q – r)` and `r ( r – p)`

Answer:`(p^2 - pq) + (q^2 - qr) + (r^2 - pr)`
`= p^2 + q^2 + r^2 - pq - qr - pr`

(b) Add: `2x (z – x – y)` and `2y (z – y – x)`

Answer: `(2xz - 2x^2- 2xy) + (2yz - 2y^2 - 2xy)`
`= 2xz - 4xy + 2yz - 2x^2 - 2y^2`

Answer: `(40ln - 12lm + 8l^2) - (3l^2 - 12lm + 15ln)`
`= 40ln - 12lm + 8l^2 - 3l^2 - 12lm + 15ln`
`= 55ln - 24lm + 5l^2`

(d) Subtract: `3a (a + b + c ) – 2 b (a – b + c)` from `4c ( – a + b + c )`

Answer:`= (-4ac + 4bc + 4c^2) - (3a^2 + 3ab + 3ac)`
`= -4ac + 4bc + 4c^2 - 3a^2 - 3ab - 3ac`
`= -7ac + 4bc + 4c^2 - 3a^2 - 3ab`