Surds

Question 1

Simplifying surds

Answer 1

√ab = √a x √b
find two numbers that x together to get the surd making sure one is a square number.

Question 2

simplify √18

Answer 2

√9 x √2 = 3√2

Question 3

simplify √200

Answer 3

√100 x √2 = 10√2

Question 4

simplify √48

Answer 4

√4 x √12 = 2√12 = 2 x √4 √3
= 2x 2 x√3
= 4 √3
however if you spot the larger square root that goes into the surd it will make the process quicker e.g √48 - √16 x √3 = 4√3

Question 5

simplify √12/√300

Answer 5

Simplify each half first.
= √4 x √3 / √100 x √3
= 2√3 / 10√3 (√3 is a common factor in numerator and denominator so they can cancel)
= 2/10 = 1/5

Question 6

√24 x √150

Answer 6

Deal with them individually first.
= √4 x √6 x √25 x √6
= 2√6 x 5√6 = 10 x 6 = 60

Question 7

√20 + √180

Answer 7

√4 x √5 + √36 x √5
= 2√5 + 6 √5 = 8 √5

Question 8

√63 - √28

Answer 8

√9 x √7 - √4 x √7 = 3√7 - 2√7 = √7

Question 9

√108 + √72

Answer 9

√36 x √3 + √36 x √2
= 6√3 + 6√2 (as the number under the square roots are different we cannot simplify any further)
however we could do -
6(√3 + √2) as there is two 6

Question 10

Expand: √3(√2 + 5)

Answer 10

√3(√2 + 5) = √6 + 5 √3

Question 11

Expand: 6(√3 + √6)

Answer 11

= 6√3 + 6 √6

Question 12

Expand: √5(8-√7)

Answer 12

8√5 - √35

Question 13

Expand: √6(√15 - 2√2)

Answer 13

= √90 - 2√12
= √9 x √10 - 2 x √4 x √3
= 3√10 - 4√3

Question 14

√12(√50 + 3√10)

Answer 14

first we can simplify the surds
2√3(5√2 + 3 √10) = 10√6 + 6√30

Question 15

Expand: (2 + √2)(3 - √5)

Answer 15

When expanding any double brackets we can use a grid.
|3 |-√5|
2| | |
√2| | |
Then we multiply the one on the side by the ones on the top.
|3 |-√5 |
2| 6 | -2 √5 |
√2| 3√2 | -√10 |
= 6 - 2√5 + 3√2 - √10

Question 16

Expand: (2 - √5)(2 + √5)

Answer 16

(Can use the use grid method, just times everything in first bracket by the other)
= -1

Question 17

Expand: (3 + √2)(2 +√3)

Answer 17

(Use the grid method)
= 6 + 2√2 + 3 √3 + √6

Question 18

Expand: (√2 + 1)(√3 - √5)

Answer 18

(use the grid method)
=√6 -√10 + √3 - √5

Question 19

Expand: (2√3 + 3√5)(2√2 - 5 √3)

Answer 19

|2√2 |-5√3
2√3| 4√6 |-10x3 = -30|
3√5 | 6√10|- 15√15 |
= 4√6 - 30 + 6√10 - 15√15

Question 20

Rationalising the Denominator

Answer 20

multiply the numerator and the denominator by the same surd(the same surd that you are trying to rationalise) We want to get rid of the surd in the denominator and place turn the numerator into a surd.

Question 21

Rationalise 2/√3

Answer 21

2/√3 x √3/√3 = 2√3/3

Question 22

Rationalise 10/√5

Answer 22

10/√5 x √5/√5 =10√5/5 = 2√5

Question 23

Rationalise
9/2√3

Answer 23

9/2√3 x √3/√3 = 9√3/6 = 3√3/2

Question 24

Rationalise 1/1 + √2

Answer 24

With questions like this when times by the surd we switch the + or - symbol to the opposite so the surds cancel each other out.
1/1+√2 x 1-√2/1-√2
= 1-√2/1-√2+√2 -2
= 1 -√2/ -1 = -1 + √2

Question 25

Rationalise 2/√2+2

Answer 25

2/√2+2 x √2-2/√2-2
= 2√2 -4/2-2√2 +2√2 -4
= 2√2 -4/-2 = -√2 + 2

Question 26

Rationalise 3/4-√5

Answer 26

3/4-√5 x 4+√5//4+√5
= 12+3√5/16+4√5-4√5-5
=12+3√5/11

Question 27

Rationalise 1+√2/3-√2

Answer 27

1+√2/3-√2 x 3+√2/3+√2
= 3+√2+3√2+2/9+3√2-3√2-2
= 5+4√2/7

Question 28

Rationalise 4+2√3/3+3√2

Answer 28

4+2√3/3+3√2 x 3-3√2/3-3√2
= 12-12√2+6√3-6√6/9-9√2+9√2-18
=12-12√2+6√3-6√6/-9
=-4+4√2-2√3+2√6/3

Question 29

Triangle ABC has base length 6√2-2√5, with area 5.
Find the exact perpendicular height of the triangle.

Answer 29

1/2 x(6√2-2√5) x h =5
(3√2 - √5) x h = 5
h = 5/3√2 - √5 x 3√2 + √5/3√2 + √5
= 15√2+5√5/18 + 3√10 - 3√10-5
=15√2+5√5/13

Question 30

Rationalise the denominator of 1/√2 + √3 - √5

Answer 30

This probably will not be in an exam but its a good tester of your surds knowledge

Question 31

solve x√2 + 5 = x - √2

Answer 31

= √2 + 5 = x - x√2
√2 + 5 = x(1 - √2)
=> x = √2 + 5/ 1 -√2 X 1+ √2/1+√2 = √2 + 2 + 5 + 5√2/ 1 - 2
= 7 + 6√2/ - 1 = -7 - 6√2

Question 32

solve √2(x-3) = 4(x + √2)

Answer 32

√2x - 3√2 = 4x + 4√2
√2x - 4x = 7√2
(√2 - 4)x = 7√2
x = 7√2/√2 - 4 x √2 + 4/√2 + 4
=14 + 28√2/2 - 16 = 14+28√2/-14
= - 1 - 2√2

Question 33

Find the exact co-ordinates of intersection of the two lines:
y = √2x + √3 - 2 and
y = √3 x + √3 - √6

Answer 33

first make one equation equal the other.
√2x + √3 - 2 = √3x + √3 - √6
(put all x’s on one side of equation)
√2x - √3x = 2 - √6
(√2 - √3)x = 2 - √6
x = 2 - √6/ √2-√3
(rationalise the denomiantor)
x= 2 - √6/ √2 - √3 x √2+√3/√2 +√3
= 2√2 + 2√3 - 2√3 - 3√2/ 2 - 3
=-√2/-1 = √2
(sub into one of the equations,equ1)
y = √2 x √2 + √3 - 2
= 2 + √3 - 2 =√3
(√2,√3)

Question 34

Prove that 1/3√2 - 4 > 2 ( > is greater then)

Answer 34

(rationalise the denominator)
1/3√2-4 x 3√2 + 4/ 3√2+4
= 3√2+4/18 - 16
= 3√2 + 4/2 = 3/2√2 + 2 > 2
as 3/2√2 is >0