1. Simplification of Surds:
Click the link Exponents to recap exponents.
Simplify the surds:
a. 4√81
b. 3√432
c. (256)3/4
d. (4√1296)/4√256
Solutions:
a. 4√81 =
4√(34) = (34)1/4 = 34 × ¼ = 3
b. 3√432 =
3√ (27 × 16) =
3√ (33 × 24) = 3√ (33 × 23 × 2) =
3√ (3 × 2)3 × 2) = 3√(3 × 2)3 × 2) =
(3√(3 × 2)3 )× ( 3√2 ) = ((3 × 2)3 )1/3 × 3√2 =
(3 × 2) × 3√2 = 6 3√2
c. (256)3/4 =
(44)3/4 = 44 × ¾ = 43 = 64
d. ( 4√196)/ (4√256)
4√(16 × 81 ) = (24 × 34 )1/4 = 24 × ¼ × 34 × ¼ = 2 × 3 = 6, and
4√256) = 4√(4)4 = (44)1/4 = 4, so,
( 4√1296)/ (4√256) = 6/4 =3/2
2. How to Add Surds:
Only like surds can be added.
Example1:
2√3 + 4√3 = 6√3
√2 and √3 cannot be added, because they are unlike surds.
Example 2:
Find the sum of √24 and √54
Solution:
Convert the given two surds into like surds as below:
√24 = √(4 × 6) = 2√6, and √54 = √(9 × 6) = 3√6
Therefore, √24 + √54 = 2√6 + 3√6 = 5√6
3. Multiplying Surds:
Example 1:
Find √12 × √48.
Solution:
√12 = √ (4 × 3) = 2 √3, and
√48 = √16 × 3 = 4√3, so,
√12 × √48 = 2 √3 × 4√3 = 8 × (√3)2 = 8 × 3 =24
Or more easily,
√12 × √48 = √12 × √(12 × 4) = √12 × √12 × √4 = 12 × 2 = 24
Example 2:
Find 51/4 × (125)0.25
Solution:
0.25 = 25/100 = 1/4
(125)0.25 = (53)1/4 = 53/4
So, 51/4 × (125)0.25 = 51/4 × (5)3/4 = 51/4 + 3/4 = 5
3. Division of Surds:
Example 1:
Divide 10 × 21/3 by 5 × 2 -2/3
Solution:
(10 × 21/3)/ (5 × 2-2/3) = 2 × 21/3 + 2/3 = 2 × 2 = 4
4. Comparison of Surds:
To compare surds, convert them into same order.
Example 1:
Which is greater: 3√4 or 4√7?
Solution:
First of all, 3√4 = 41/3 , and 4√7 = 71/4
Now, convert both surds into same order as below:
Raise each surd to power 12, which is L.C.M. of 3 and 4, the order of the two surds.
41/3 = (41/3)12 = 44 = 256, and
71/4 = (71/4)12 = 73 = 343
Since 343 > 256, so 4√7 > 3√4
Example 2:
Arrange the following surds in ascending order:
3√9, 4√11 and 6√17
Solution:
Write the given surds as follows:
91/3, 111/4 , 171/6
The surds are of order 3, 4 and 6 respectively.
Find L.C.M. of 3, 4 and 6, which is 12.
Now, raise each surd to power 12.
(91/3)12 = 912/3 = 94 = 6561,
(111/4 )12 = 1112/4 = 113 = 1331
(171/6)12 = 1712/6 = 172 = 289
Clearly we see that 91/3 > 111/4 > 171/6
Example 3:
Which is greater: √10 – √8 or √11 – √9?
Solution:
To compare, square both the surds.
(√10 – √8)2 = 10 – 8 – 2√10 × √8 = 2 – 2√80, and
(√11 – √9) = 11 – 9 – 2 √11 × √9 = 2 – 2 √99
Now, compare 2 – 2√80 and 2 – 2√99 as below:
Assume:
2 – 2√80 > 2 – 2√99
Strike out the common number 2 .
2 – 2√80 > 2 – 2√99,
So, we have:
– 2√80 > – 2√99,
Strike out the common factor – 2, and reverse the inequality
– 2√80 > – 2√99, we get: 80 < 99 which is true.
Since 80 < 99 is true, therefore our assumption 2 – 2√80 > 2 – 2√99
is also true.
Therefore,
√10 – √8 > √11 – √9
5. Rationalization of Surds
Surds are often rationalized.
Converting surds which are irrational numbers into a rational number is called rationalization.
6. What is a Rationalizing factor?
If the product of two surds is a rational number, then each factor is a rationalizing factor of the other.
How to rationalize √2?
Multiply it to itself:
√2 × √2 = 2, a rational number.
So, √2 is called the rationalizing factor of √2.
Example 2:
What is the rationalizing factor of √2 – 1?
Solution:
Multiply √2 + 1 to √2 – 1 to make √2 – 1 a rational number.
(√2 – 1) × (√2 + 1) = 2 – 1 = 1
So, √2 + 1 is the Rationalizing factor of √2 – 1
Example 3:
Make the denominator rational in:
1/ (√2 - √3)
Solution:
To rationalize denominator √2 - √3 is to make it free of square roots.
Multiply √2 - √3 with its conjugate surd √2 + √3
(√2 - √3) × (√2 + √3) = 2 – 3 = – 1
Now, the method of Rationalization is like this:
1/ (√2 - √3) =
[1/ (√2 - √3)] × [(√2 + √3)/ (√2 + √3)] =
(√2 + √3)/ [(√2 – √3)/ (√2 + √3)] =
(√2 + √3)/(- 1) = -(√2 + √3)
The denominator is rationalized (made free of surds)
Example 4:
What is the rationalizing factor of 21/3 + 2-1/3 ?
Solution:
The rationalizing factor of a1/3 + b-1/3 is:
a2/3 + b-2/3 - ab, because
(a1/3 + b-1/3 ) × (a2/3 + b-2/3 – ab) = a – b
Therefore, the rationalizing factor of 21/3 + 2-1/3 is :
22/3 + 2-2/3 – 1
Important Formulas on Rationalization Factors of Surds:
What is the Surd? |
What is its Rationalizing Factor? |
√a |
√a |
√a + √b |
√a – √b |
1/√a |
√a |
1/(√a + √b) |
√a – √b |
1/(√a + √b) |
√a – √b |
How to find square root of surds:
Example 1:
Let us find the square root of 5 + 2√6.
Think of two numbers a and b, such that:
a + b = 5 and a × b = 6.
They are: 2 + 3 = 5 and 2 × 3 = 6
Therefore, 5 + 2√6 = (2 + 3) + 2 × (√2) × (√3)
Now, express 2 as (√2)2 and 3 as (√3)2
Therefore,
5 + 2√6 = [(√2)2 + (√3)2] + 2 × (√2) × (√3) =
[(√2)2 + (√3)2]2
√ (5 + 2√6)2 = √[(√2)2 + (√3)2]2 = √2 + √3.