Surds – Exercise 7.1
- Simplify the following surds:
(i) √75 + √108 – √192
(ii) 4√12 – √50 – 7√48
(iii) √45 – 3√20 + 3√5
(iv) 2√(2a) + 3√(8a) – √(2a)
(v) 3x√x + 3√x3 – 2√(9x3)
(vi) √12 + √50 +5√3 – √147 – √32
(vii) 4√7 – 3√252 + 5√343
(viii) 1/8 √50 + 1/6√75 – 1/8 √18 – 1/3√3
- Find sum of the following surds
(i) x√y , 2x√y, 4x√y
(ii) 5∛p, 3∛p, 2∛p
(iii) x√x, y√y, 3√x3, 4√y3
(iv) (√12+20), (3√3 + 2√5), (√45 – √90)
(v) (√3 + √2), (2√2 + 3√3), (4√2 – 3√3)
(vi) (√x +2√y), (2√x – 3√y), (3√x + √y)
III.
- Subtract 5√x from 9√x ad express the result in index form.
- Subtract 3√p from 10√p
- Subtract 3√a from the sum of 4√a and 2√a
- Subtract 2√x + 3√y from 5√x – √y
Surds – Exercise 7.1 – Solution:
- Simplify the following surds:
(i) √75 + √108 – √192
Solution:
√75 + √108 – √192
= √(3x 25) + √(3×36) – √(3×64)
= 5√3 + 6√3 – 8√3
= (5 + 6 – 8) √3
= 3√3
(ii) 4√12 – √50 – 7√48
Solution:
4√12 – √50 – 7√48
= 4√(4×3) – √(25×2) – 7√(16×3)
= 4×2√3 – 5√2 – 7×4√3
= 8√3 – 5√2 – 28√3
= (8 – 28)√3 – 5√2
= – 20√3 – 5√2
= -5(4√3 + √2)
(iii) √45 – 3√20 + 3√5
Solution:
√45 – 3√20 + 3√5
= √(9×5) – 3√(4×5) + 3√5
= 3√5 – 3×2√5 + 3√5
= 3√5 – 6√5 + 3√5
= 0
(iv) 2√(2a) + 3√(8a) – √(2a)
Solution:
= 2√(2a) + 3√(8a) – √(2a)
= 2√(2a) + 3√(4x2a) – √(2a)
= 2√(2a) + 3×2√(2a) – √(2a)
= (2 + 6 – 1)√(2a)
= 7√(2a)
(v) 3x√x + 3√x3 – 2√(9x3)
Solution:
3x√x + 3√x3 – 2√(9x3)
= 3x√x + 3√(x*x2) – 2√(x*9x2)
= 3x√x + 3x√x – 2*3x√x
= 3x√x + 3x√x – 6x√x
= (3x + 3x – 6x)√x
= 0
(vi) √12 + √50 +5√3 – √147 – √32
Solution:
√12 + √50 +5√3 – √147 – √32
= √(4×3) + √(25×2) + 5√3 – √(49×3) – √(16×2)
= 2√3 + 5√2 + 5√3 – 7√3 – 4√2
= (2 + 5 – 7)√3 +(5 – 4)√2
= 0. √3 + √2
= √2
(vii) 4√7 – 3√252 + 5√343
Solution:
4√7 – 3√252 + 5√343
= 4√7 – 3√(7×36) + 5√(7×49)
= 4√7 – 3×6√7 + 5×7√7
= (4 – 18 + 35) √7
= 21√7
(viii) 1/8 √50 + 1/6√75 – 1/8 √18 – 1/3√3
Solution:
1/8 √50 + 1/6√75 – 1/8 √18 – 1/3√3
= 1/8 √(2×25) + 1/6√(25×3) – 1/8 √(9×2) – 1/3√3
= 1/8 x 5√2 + 1/6 x 5√3 – 1/8 x 3√2 – 1/3√3
= (5/8 – 3/8)√2 + (5/6 – 1/3) √3
=1/4 √2 + 1/2 √3
2. Find sum of the following surds
(i) x√y , 2x√y, 4x√y
Solution:
x√y + 2x√y + 4x√y = (x + 2x + 4x)√y
= 7x√y
(ii) 5∛p, 3∛p, 2∛p
Solution:
5∛p + 3∛p + 2∛p = (5 + 3+ 2) ∛p
= 10∛p
(iii) x√x, y√y, 3√x3, 4√y3
Solution:
x√x + y√y + 3√x3 + 4√y3
= x√x + y√y + 3√(x*x2) + 4√(y*y2)
= (x + 3x)√x + (y + 4y) √y
= 4x√x + 5y√y
(iv) (√12+√20), (3√3 + 2√5), (√45 – √90)
Solution:
= √12+√20 + 3√3 + 2√5 + √45 – √90
= √(4×3) + √(4×5) + 3√3 + 2√5 + √(9×5) – √(9×10)
= 2√3 + 2√5 + 3√3 + 2√5 + 3√5 – 3√10
= (2+3)√3 + (2+2+3)√5 – 3√10
= 5√3 + 7√5 – 3√10
(v) (√3 + √2), (2√2 + 3√3), (4√2 – 3√3)
Solution:
= √3 + √2 + 2√2 + 3√3 + 4√2 – 3√3
= (1+ 3 – 3)√3 + (1+2+4)√2
= √3 + 7√2
(vi) (√x +2√y), (2√x – 3√y), (3√x + √y)
Solution:
= √x +2√y + 2√x – 3√y + 3√x + √y
= (1+2 + 3)√x +(2 – 3 + 1)√y
= 6√x + 0. √y
= 6√x
III.
- Subtract 5√x from 9√x ad express the result in index form.
Solution:
9√x – 5√x = (9 – 5) √x = 4√x
Therefore, 4√x is the result when 5√x subtracted from 9√x.
Then the index form of the result 4√x = 4x1/2
- Subtract 3√p from 10√p
Solution:
10√p – 3√p = (10 – 3)√p = 7√p
- Subtract 3√a from the sum of 4√a and 2√a
Solution:
We need to find the sum of 4√a and 2√a:
4√a + 2√a = (4+2)√a + 6√a
We have to subtract 3√a from the sum of 4√a and 2√a:
6√a + 3√a = (6+3)√a = 9√a
- Subtract 2√x + 3√y from 5√x – √y
Solution:
(5√x – √y) – (2√x + 3√y)
= 5√x – √y – 2√x – 3√y
= (5 – 2)√x + (-1 – 3)√y
= 7√x – 4√y
Next exercise – Surds – Exercise 7.2 – Class X