Next exercise – Surds – Exercise 7.2 – Class X
Surds – Exercise 7.3
- Write the rationalizing factor for the following surds.
1) √a
2) 2√x
3) 7√y
4) √(xy)
5) 4√(x + y)
6) 8√(x – y)
7) 1/2√p
8) a√(ab)
9) x√(mn)
10)5p√(a+b)
- Write the conjugate of the following binomial surds
(i) √a + √b
(ii) √x – 2√y
(iii) 3√p – 2√q
(iv) x + 3√y
(v) 10√2 + 3√5
(vi) 5 + √3
(vii) √8 – 5
(viii) 3√7 + 7√3
(ix) 1/2 + √2
(x) 1/2 x + 1/2√y
(xi) x√a + y√b
(xii) xy√z + yz√x
III. Find the rationalising factor of the following binomial surds
(i) 21/3 + 21/3
(ii) 51/3 + 51/3
(iii) √(1+y) – √(1-y)
Surds – Exercise 7.3
- Write the rationalizing factor for the following surds.
1) √a
Solution:
√a x √a = a
Therefore, rationalizing factor of √a is √a
2) 2√x
Solution:
2√x x √x = 2x
Therefore, rationalizing factor of 2√x is √x
3) 7√y
Solution:
7√y x √y = 7y
Therefore, rationalizing factor of 7√y is √y
4) √(xy)
Solution:
√(xy) x √(xy) = xy
Therefore, rationalizing factor of √(xy) is √xy
5) 4√(x + y)
Solution:
4√(x+y) x √(x+y) = 4(x+y)
Therefore, rationalizing factor of 4√(x+y) is √(x+y)
6) 8√(x – y)
Solution:
8√(x – y) x √(x – y) = 8(x – y)
Therefore, rationalizing factor of 8√(x – y) is √(x-y)
7) 1/2√p
Solution:
1/2√p x √p = 1/2 p
Therefore, rationalizing factor of 1/2√p is √p
8) a√(ab)
Solution:
a√(ab) x √(ab) = a(ab) = a2b
Therefore, rationalizing factor of a√(ab) is √(ab)
9) x√(mn)
Solution:
x√(mn) x √(mn) = xmn
Therefore, rationalizing factor of x√(mn) is xmn
10)5p√(a+b)
Solution:
5p√(a+b) x √(a+b) = 5p(a+b)
Therefore, rationalizing factor of 5p√(a+b) is √(a+b)
II. Write the conjugate of the following binomial surds
(i) √a + √b
Solution:
The conjugate of binomial surd √a + √b is √a – √b
(ii) √x – 2√y
Solution:
The conjugate of binomial surd √x – 2√y is √x + 2√y
(iii) 3√p – 2√q
Solution:
The conjugate of binomial surd 3√p – 2√q is 3√p + 2√q
(iv) x + 3√y
Solution:
The conjugate of binomial surd x + 3√y is x – 3√y
(v) 10√2 + 3√5
Solution:
The conjugate of binomial surd 10√2 + 3√5 is 10√2 – 3√5
(vi) 5 + √3
Solution:
The conjugate of binomial surd 5 + √3 is 5 – √3
(vii) √8 – 5
Solution:
The conjugate of binomial surd √8 – 5 is √8 + 5
(viii) 3√7 + 7√3
Solution:
The conjugate of binomial surd 3√7 + 7√3 is 3√7 – 7√3
(ix) 1/2 + √2
Solution:
The conjugate of binomial surd 1/2 + √2 is 1/2 – √2
(x) 1/2 x + 1/2√y
Solution:
The conjugate of binomial surd 1/2 x + 1/2√y is 1/2 x – 1/2√y
(xi) x√a + y√b
Solution:
The conjugate of binomial surd x√a + y√b is x√a – y√b
(xii) xy√z + yz√x
Solution:
The conjugate of binomial surd xy√z + yz√x is xy√z – yz√x
III. Find the rationalizing factor of the following binomial surds
(i) 21/3 + 2 -1/3
Solution:
21/3 + 2-1/3
a = 21/3 ; b = 2 -1/3
Then, a3 = (21/3)3 = 2 and b3 = (2-1/3)3 = 1/2
a3 + b3 = 2 + 1/2 = 4+1/2 = 5/2
But a3 + b3 = (a + b)(a2 – ab + b2)
5/2 = (21/3 + 2-1/3)[(21/3)2 – 21/32-1/3 + (2-1/3)2]
5/2 = (21/3 + 2-1/3)(22/3 + 2-2/3 – 1)
Since 3/2 is a rational number, rationalizing factor of (21/3 – 2-1/3) is (22/3 + 2-2/3 + 1)
(ii) 51/3 + 5-1/3
Solution:
51/3 + 5-1/3
a = 51/3 ; b = 5 -1/3
Then, a3 = (51/3)3 = 5 and b3 = (5-1/3)3 = 1/5
a3 + b3 = 5 + 1/5 = 25+1/5 = 26/5
But a3 + b3 = (a + b)(a2 – ab + b2)
26/5 = (51/3 + 5-1/3)[(51/3)2 – 51/35-1/3 + (5-1/3)2]
26/5 = (51/3 + 5-1/3)(52/3 + 5-2/3 – 1)
Since 24/5 is a rational number, rationalizing factor of (51/3 + 5-1/3) is (52/3 + 5-2/3 – 1)
(iii) √(1+y) – √(1-y)
Solution:
[√(1+y) – √(1-y)]x[√(1+y) + √(1-y)] =
= [√(1+y)]2 – [√(1-y)]2
= (1+ y) – (1 – y)
= 1 + y – 1 + y
= 2y
⸫√(1+y) – √(1-y) is rationalised by using its conjugate √(1+y) + √(1-y).
Therefore, rationalizing factor of √(1+y) – √(1-y) is √(1+y) + √(1-y).
Next Exercise – Surds – Exercise 7.4 – Class X