1. Rationalizing surds means removing the radical sign from the denominator by multiplying the numerator and denominator by the conjugate. 2. The conjugate of a surd term keeps the radicand the same but changes the sign of any terms outside the radical. 3. Rationalizing terms of the form a - b involves multiplying the numerator and denominator by the conjugate a + b.

1. Simplification of Surds
To simplify surds we do one of the
following :
1. Split the numerator
2. Expand brackets
3. Rationalize
4. Apply the properties

2. Rationalization of Surds
𝑎
𝑏
=
𝑎
𝑏
×
𝑏
𝑏
=
𝑎𝑏
𝑏

3. Rationalization of Surds
𝑎
1 − 𝑏
=
𝑎
1 − 𝑏
×
1 + 𝑏
1 + 𝑏
=
𝑎(1 + 𝑏)
(1 − 𝑏)(1 + 𝑏)

4. Rationalization of Surds
𝑎
𝑎 − 𝑏
=
𝑎
𝑎 − 𝑏
×
𝑎 + 𝑏
𝑎 + 𝑏
=
𝑎( 𝑎 + 𝑏)
(1 − 𝑏)(1 + 𝑏)

5. JAMB 2010
Rationalize
2 3+ 5
5− 3
Solution
2 3 + 5
5 − 3
×
5 + 3
5 + 3
=
2 3( 5 + 3) + 5 5 + 3
5
2
− 3
2
=
2 15 + 2 3 + 5 + 15
2
=
3
2
15 + 11

6. WASSCE May/June 2014
Simplify 3 75 − 12 + 108
leaving your answer in surd form.
Solution
3 75 − 12 + 108
= 3 25 × 3 − 4 × 3 + 36 × 3
= 3 25 × 3 − 4 × 3 + 36 × 3
= 3 × 5 × 3 − 2 × 3 + 6 × 3
= 15 3 − 2 3 + 6 3
= 19 3

7. WASSCE May/June 2013
Simplify
3
4
128 − 50 leaving your
answer in surd form.
Solution
3
4
128 − 50
=
3
4
64 × 2 − 25 × 2
=
3
4
× 8 2 − 5 2
= 𝟐

8. SSSCE Nov. 2004
Without using Calculator, evaluate;
3 7 7 − 2 7 , If 7 = 2.646
Solution
3 7 7 − 2 7
= 𝟐𝟏 𝟕 − 𝟔 × 𝟕
= 𝟐𝟏 𝟕 − 𝟒𝟐
𝒊𝒇 7 = 2.646,
= 𝟐𝟏(𝟐. 𝟔𝟒𝟔) − 𝟒𝟐
= 𝟏𝟑. 𝟓𝟔𝟔

9. 1. Rationalize
2
2
A.
2
2
B. 2 2
C. 2
D. − 2
E.
1
2
F. The correct answer is 2
𝑆𝑜𝑙𝑢𝑡𝑖𝑜𝑛
2
2
=
2
2
×
2
2
=
2 2
2
= √2

10. 2. Simplify √3( 3 − 2)
A. 3 + 6
B. 36 − 6
C. 9 − 6
D. 6 − 6
E. 3 − 6
The correct answer is 3 − 6
Solution
3( 3 − 2)
= 3
2
− 3 × 2
= 3 − 6

11. 3. Given that 28 − 63 = 𝑚 7. Find the value of 𝑚.
A. −4
B. −3
C. −2
D. −1
E. 1
The correct answer is −1
Solution
28 − 63 = 𝑚 7
4 × 7 − 9 × 7 = 𝑚 7
2 7 − 3 7 = 𝑚 7
− 7 = 𝑚 7
𝑚 = −1

12. 4. Given that 2 = 𝑝 and 7 = 𝑞. Express 3.5 in terms of 𝑝 𝑎𝑛𝑑 𝑞.
A.
𝑝
𝑞
B.
𝑝
2𝑞
C.
1
2
𝑝𝑞
D.
2𝑝
𝑞
E. 𝑝 − 𝑞
The correct answer is
1
2
𝑝𝑞
Solution
3.5 =
7
2
=
7
2
=
7
2
×
2
2
=
7 × 2
2
=
1
2
𝑝𝑞

13. 5. Simplify
6− 42
7
A.
6( 7+1)
7
B.
6( 7−6)
7
C.
6( 7−√6)
7
D.
6(1− 7)
7
E.
6( 7−1)
7
The correct answer is
6( 7−1)
7
Solution
6 − 42
7
=
6
7
−
42
7
=
6
7
−
42
7
=
6
7
− 6
=
6
7
×
7
7
− 6
=
6 × 7
7
− 6
=
6( 7 − 1)
7

14. 6. Given that 𝑝 = 3 and 𝑞 = 7. Evaluate (𝑝 − 𝑞)(𝑝 + 𝑞)
A. −2
B. 3
C. −4
D. 6
E. 5
The correct answer is −4
Solution
𝑝 − 𝑞 𝑝 + 𝑞 = 𝑝2
− 𝑞2
= 3
2
− 7
2
= 3 − 7
= −4

15. 7. Given that 𝑝 = 3 and 𝑞 = 7. Evaluate
3𝑝
7𝑞
A.
3 21
19
B.
3 21
21
C.
3 11
49
D.
3 21
49
E.
3 21
10
The correct answer is
3 21
49
Solution
3𝑝
7𝑞
=
3 3
7 7
=
3 3
7 7
×
7
7
=
3 21
7 × 7
=
3 21
49

16. 8. If 2𝑥 =
1
2
. Find the value of 𝑥
A. 𝑥 =
2
4
B. 𝑥 =
2
2
C. 𝑥 =
2
8
D. 𝑥 =
2
16
E. 𝑥 =
2
32
The correct answer is𝑥 =
2
4
Solution
2𝑥 =
1
2
Dividing through by 2
We have
𝑥 =
1
2 2
𝑥 =
1
2 2
×
2
2
𝑥 =
2
2(2)
𝑥 =
2
4

17. 9. If 2𝑥 − 2𝑥 = √6. Find the value of 𝑥
A. 𝑥 =
6( 2+2)
2
B. 𝑥 = −
6( 2+2)
2
C. 𝑥 = −
6( 2−2)
2
D. 𝑥 =
6( 2−2)
2
E. 𝑥 =
6( 2−2)
4
The correct answer is𝑥 = −
6( 2+2)
2
Solution
2𝑥 − 2𝑥 = √6
Factor 𝑥 to obtain;
2 − 2 𝑥 = √6
Dividing through by 2 − 2
We have
𝑥 =
√6
2 − 2
Rationalize ;
𝑥 =
√6
2 − 2
×
2 + 2
2 + 2
𝑥 =
6( 2 + 2)
−2
𝑥 = −
6( 2 + 2)
2

18. 10. Simplify
3
7 3
A.
3
7
B.
3
3
C.
3
21
D.
3
17
E.
3 3
7
The correct answer is
3
7
Solution
=
3
7 3
×
3
3
=
3 3
7(3)
=
3
7

19. SUMMARY
Rationalizing surds means removing the radical sign from the denominator
The conjugate of 𝑎 is still 𝑎
The conjugate of 𝑎 − 𝑏 is still 𝑎 + 𝑏