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Example 7: Find the volume of a cuboid if its length is a b units, breadth is b c units, and height
2
is c a units.
Solution: Volume of a cuboid Quick Check
= length × breadth × height Find the volume of each rectangular
boxes with given length, breadth
2
= (a b × b c × c a) cubic units and height in the following table:
2
2
= (a 2+1 ) × (b 1+2 ) × (c 1+2 ) cubic units Length Breadth Height Volume
3 3 3
= a b c cubic units 7kl 5l 2 5 6k 2
2 3
7 2
3a b 4ab a b
2
2 2
Example 8: Find the product of m n , 2mn, and m n; 6
2
and then evaluate it for m = 2 and n = –1. 20x y 15y 2 7x 4
Solution: m n × 2mn × m n = 2(m × m × m ) × (n × n × n) = 2m × n 4
2
2
2
2 2
5
2
For m = 2 and n = –1, we have
5
4
5
4
2m × n = 2(2) × (–1) = 64 × 1 = 64
Practice Time 8B
1. Find the product of the following.
2
2
2
(a) 7xy and –4xy (b) –6ab and 3 a bc (c) –5a bc, 11ab and 13abc 2
3 2
3 3
3
3
(d) a bc , a b c and 9abc (e) (–16pq), (20p q ), (15p) and (–13q)
2 2
2
4 3
2. Simplify the following and also find the value of the product for x = 1, y = –1 and z = 2.
2
3 5
2
2 3
2
2
(a) 19x yz × 7xy z × 5xyz (b) –7x y × –9xyz × 17x z × (–y z)
3. Find the area of the rectangle with the following length and breadth respectively.
 1   1 3 2 
2
2
3
2
2
(a)   2pq, 4 pqr   (b) (1.5x y, 2.5x ) (c)   4 xy , 7 yz  
4. Add the product of (–3ab ) and (1.7ab) to the product of (5a b ) and (–2ab).
2 3
2
Multiplication of a Monomial by a Binomial
Let us now multiply a monomial, say ‘a’ with a binomial, say (b + c), we follow these steps:
Step 1: Write the product of monomial and binomial using the multiplication symbol, i.e., a × (b + c)
or a × (b – c).
Step 2: Use the Distributive Law: First, multiply the monomial by the first term of the binomial.
And then multiply the monomial by the second term of the binomial i.e., a × (b + c) = a × b + a × c
or a × (b – c) = a × b – a × c.
Step 3: Simplify the terms, i.e., ab + ac or ab – ac.
Step 4: If there are like terms in the product then combine them.
Example 9: Multiply the following algebraic expressions:
Think and Answer
(a) 4x × (3x + 6y) (b) –3x and (7x – 2)
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Solution: (a) 4x × (3x + 6y) = 4x(3x) + 4x(6y) = 12x + 24xy Suppose the cost of planting one
flower is `10 and you have (2x + 5)
(b) –3x × (7x – 2) = (–3x) × (7x) + (–3x) × (–2) flowers to plant. Write an expression
= –21x + 6x for the total cost.
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Mathematics-8 200