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Simplifying Surds (5)

In this worksheet, students manipulate and simplify expressions involving surds.

'Simplifying Surds (5)' worksheet

Key stage:  KS 4

Curriculum topic:  Algebra

Curriculum subtopic:  Simplify/Manipulate Algebraic Expressions/Fractions

Difficulty level:  

down

Worksheet Overview

QUESTION 1 of 10

A surd is a root that is irrational.

Examples of surds include √a, √2, 3√11, √a + √2, 3√15

 

Use the following rules to simplify surds:

ab = √a x √b

√a/√b = √(a/b)

Remember that √a + b ≠ √a + √b

 

Example

Simplify the following as a surd in the form a + b√c, where c is an integer and is as small as possible, or if the answer is a whole number, just state the whole number.

 

(3 + √5)(3 - 2√5)
45


Answer

 First we simplify the numerator remembering that √5 x √5 = 5

 

(3 + √5)(3 - 2√5)

 

= 3(3 - 2√5) + 5(3 - 2√5)

 

= 9 - 6√5 + 3 - 2 x 5

 

= + 3 - 10 - 6√5

 

= 2 - 6√5

 

So we now have

 

2 - 6√5
45

which is equal to

 

2 - 6√5

9 x 5

 

which is equal to

2 - 6√5
3√5

 

 

multiply top and bottom by √5 to get

 

25 - 6 x 5
3 x 5

 

which simplifies to

25 - 30
15


which equals

25
   -   2
15


 

Simplify the following as a surd in the form a + b√c, where c is an integer and is as small as possible, or if the answer is a whole number, just state the whole number.

 

(1 + √5)(1 + √5)

1 +3√5

6 + 2√5

-1 + 3√5

Simplify the following as a surd in the form a + b√c, where c is an integer and is as small as possible, or if the answer is a whole number, just state the whole number.

 

(3 + 2√5)(3 - √5)

19 +3√5

10√5

-1 + 3√5

Simplify the following as a surd in the form a + b√c, where c is an integer and is as small as possible, or if the answer is a whole number, just state the whole number.

 

(1 + √5)(1 - √5)

-4

6 + 2√5

-1 + 3√5

Simplify the following as a surd in the form a + b√c, where c is an integer and is as small as possible, or if the answer is a whole number, just state the whole number.

 

(1 + 2√3)(6 - √3)

19 +3√3

9√3

-1 + 3√3

Simplify the following as a surd in the form a + b√c, where c is an integer and is as small as possible, or if the answer is a whole number, just state the whole number.

  

(3 + √5)(3 - 2√5)
2

 

19 +3√5

9√5

1 - 3√5

Simplify the following as a surd in the form a + b√c, where c is an integer and is as small as possible, or if the answer is a whole number, just state the whole number.

 

(1 + 2√3)2

13 +4√3

13√3

37 + 4√3

Simplify the following as a surd in the form a + b√c, where c is an integer and is as small as possible, or if the answer is a whole number, just state the whole number.

  

(4 + 3√2)(2 - √2)
2

 

12 +3√2

6√2

1 + √2

Simplify the following as a surd in the form a + b√c, where c is an integer and is as small as possible, or if the answer is a whole number, just state the whole number.

  

45 - √50
5

 

15 - √10

15√10

9 - 2√5

Simplify the following as a surd in the form a + b√c, where c is an integer and is as small as possible, or if the answer is a whole number, just state the whole number.

 

(3 + 2√5)(3 - 2√5)

19 +3√5

-11

-1 + 3√5

Simplify the following as a surd in the form a + b√c, where c is an integer and is as small as possible, or if the answer is a whole number, just state the whole number.

 

(3 + 2√7)2

13 +4√7

24√7

37 + 12√7

  • Question 1

Simplify the following as a surd in the form a + b√c, where c is an integer and is as small as possible, or if the answer is a whole number, just state the whole number.

 

(1 + √5)(1 + √5)

CORRECT ANSWER
6 + 2√5
EDDIE SAYS
1(1 + √5) + √5(1 + √5)
  • Question 2

Simplify the following as a surd in the form a + b√c, where c is an integer and is as small as possible, or if the answer is a whole number, just state the whole number.

 

(3 + 2√5)(3 - √5)

CORRECT ANSWER
-1 + 3√5
EDDIE SAYS
3(3 - √5) + 2√5(3 - √5)
  • Question 3

Simplify the following as a surd in the form a + b√c, where c is an integer and is as small as possible, or if the answer is a whole number, just state the whole number.

 

(1 + √5)(1 - √5)

CORRECT ANSWER
-4
EDDIE SAYS
1(1 - √5) + √5(1 - √5)
  • Question 4

Simplify the following as a surd in the form a + b√c, where c is an integer and is as small as possible, or if the answer is a whole number, just state the whole number.

 

(1 + 2√3)(6 - √3)

CORRECT ANSWER
9√3
EDDIE SAYS
1(6 - √3) + 2√3(6 - √3)
  • Question 5

Simplify the following as a surd in the form a + b√c, where c is an integer and is as small as possible, or if the answer is a whole number, just state the whole number.

  

(3 + √5)(3 - 2√5)
2

 

CORRECT ANSWER
1 - 3√5
EDDIE SAYS
3(3 - 2√5) + √5(3 - 2√5)
then divide by 2
  • Question 6

Simplify the following as a surd in the form a + b√c, where c is an integer and is as small as possible, or if the answer is a whole number, just state the whole number.

 

(1 + 2√3)2

CORRECT ANSWER
13 +4√3
EDDIE SAYS
1(1 + 2√3) + 2√3(1 + 2√3)
  • Question 7

Simplify the following as a surd in the form a + b√c, where c is an integer and is as small as possible, or if the answer is a whole number, just state the whole number.

  

(4 + 3√2)(2 - √2)
2

 

CORRECT ANSWER
1 + √2
EDDIE SAYS
4(2 - √2) + 3√2(2 - √2)
then divide by 2
  • Question 8

Simplify the following as a surd in the form a + b√c, where c is an integer and is as small as possible, or if the answer is a whole number, just state the whole number.

  

45 - √50
5

 

CORRECT ANSWER
15 - √10
EDDIE SAYS
Multiply top and bottom by √5
then simplify
  • Question 9

Simplify the following as a surd in the form a + b√c, where c is an integer and is as small as possible, or if the answer is a whole number, just state the whole number.

 

(3 + 2√5)(3 - 2√5)

CORRECT ANSWER
-11
EDDIE SAYS
3(3 - 2√5) + 2√5(3 - 2√5)
  • Question 10

Simplify the following as a surd in the form a + b√c, where c is an integer and is as small as possible, or if the answer is a whole number, just state the whole number.

 

(3 + 2√7)2

CORRECT ANSWER
37 + 12√7
EDDIE SAYS
3(3 + 2√7) + 2√7(3 + 2√7)
---- OR ----

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