basic_engineering_mathematics0

Algebra 41
Using the third and fourth laws of indices gives:
(x 2 y 1/2 )( √ √
x 3 y2 )
= (x2 y 1/2 )(x 1/2 y 2/3 )
(x 5 y 3 ) 1/2 x 5/2 y 3/2
Using the first and second laws of indices gives:
x 2+(1/2)−(5/2) y (1/2)+(2/3)−(3/2) = x 0 y −1/3
= y −1/3 or
1
y 1/3
from the fifth and sixth laws of indices.
or
1
3√ y
Problem 26. Remove the brackets and simplify the expression
(3a + b) + 2(b + c) − 4(c + d)
Both b and c in the second bracket have to be multiplied by 2,
and c and d in the third bracket by −4 when the brackets are
removed. Thus:
(3a + b) + 2(b + c) − 4(c + d)
= 3a + b + 2b + 2c − 4c − 4d
Collecting similar terms together gives: 3a + 3b − 2c − 4d
Now try the following exercise
Exercise 22
Further problems on laws of indices
(Answers on page 272)
1. Simplify (x 2 y 3 z)(x 3 yz 2 ) and evaluate when x = 1 2 , y = 2
and z = 3
2. Simplify (a 3/2 bc −3 )(a 1/2 b −1/2 c) and evaluate when
a = 3, b = 4 and c = 2
3. Simplify a5 bc 3
a 2 b 3 c 2 and evaluate when a = 3 2 , b = 1 2 and
c = 2 3
In Problems 4 to 11, simplify the given expressions:
4.
6.
8.
x 1/5 y 1/2 z 1/3
x −1/2 y 1/3 z −1/6 5.
p 3 q 2
pq 2 − p 2 q
a 2 b + a 3 b
a 2 b 2
7. (a 2 ) 1/2 (b 2 ) 3 (c 1/2 ) 3
(abc) 2
(a 2 b −1 c −3 ) 3 9. ( √ x √ y 3 3 √
z2 )( √ x √ y 3√ z 3 )
10. (e 2 f 3 )(e −3 f −5 ), expressing the answer with positive
indices only
11.
(a 3 b 1/2 c −1/2 )(ab) 1/3
( √ a 3√ bc)
6.3 Brackets and factorization
When two or more terms in an algebraic expression contain
a common factor, then this factor can be shown outside of
a bracket. For example
ab + ac = a(b + c)
which is simply the reverse of law (v) of algebra on page 36, and
6px + 2py − 4pz = 2p(3x + y − 2z)
This process is called factorization.
Problem 27. Simplify a 2 − (2a − ab) − a(3b + a)
When the brackets are removed, both 2a and −ab in the first
bracket must be multiplied by −1 and both 3b and a in the second
bracket by −a. Thus
a 2 − (2a − ab) − a(3b + a)
= a 2 − 2a + ab − 3ab − a 2
Collecting similar terms together gives: −2a − 2ab.
Since −2a is a common factor, the answer can be expressed as
−2a(1 + b)
Problem 28. Simplify (a + b)(a − b)
Each term in the second bracket has to be multiplied by each
term in the first bracket. Thus:
(a + b)(a − b) = a(a − b) + b(a − b)
= a 2 − ab + ab − b 2 = a 2 − b 2
Alternatively
a + b
a − b
Multiplying by a → a 2 + ab
Multiplying by −b → −ab − b 2
Adding gives: a 2 − b 2
Problem 29. Remove the brackets from the expression
(x − 2y)(3x + y 2 )
(x − 2y)(3x + y 2 ) = x(3x + y 2 ) − 2y(3x + y 2 )
= 3x 2 + xy 2 − 6xy − 2y 3
Problem 30. Simplify (2x − 3y) 2