 # Simplifying Surds (4)

In this worksheet, students manipulate and simplify expressions involving surds. Key stage:  KS 4

Curriculum topic:  Algebra

Curriculum subtopic:  Simplify/Manipulate Algebraic Expressions/Fractions

Difficulty level:   ### 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, where a and b are integers and b is as small as possible, or if the answer is a whole number, just state the whole number.

√108 - √12

Look for a low common factor of 108 and 12, which leaves square numbers when factorised.  Here 3 is a common factor that leaves 36 and 4 respectively.

Thus:

108 = 36 x 3

12 = 4 x 3

√108 - √12

= 36 x 3 - √4 x 3

= √36 x √3 - √4 x √3

= 6 x √3 - 2 x √3

= (6 - 2) x √3

= 4√3

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

√12 + √48

4√2

10√18

6√3

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

√45 + √125

2√40

7√5

8√5

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

√27 + √12

√39

5√3

6√3

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

√27 - √12

√3

5√3

6√3

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

√80 - √20

2√5

7√5

√60

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

√27 + √48

4√2

10√18

7√3

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

√50 + √98

12√2

10√18

7√48

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

√63 - √28

√7

5√35

6√3

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

√300 + √12

√312

5√32

12√3

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

√275 - √44

3√11

5√35

6√22

• Question 1

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

√12 + √48

6√3
EDDIE SAYS
12 = 4 × 3
48 = 16 × 3
4 × 3 + √16 × 3 = √4 x √3 + √16 × √3
= 2√3 + 4√3
• Question 2

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

√45 + √125

8√5
EDDIE SAYS
45 = 9 × 5
125 = 25 × 5
9 × 5 + √25 × 5 = √9 x √5 + √25 × √5
= 3√5 + 5√5
• Question 3

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

√27 + √12

5√3
EDDIE SAYS
27 = 9 × 3
12 = 4 × 3
9 × 3 + √4 × 3 = √9 x √3 + √4 × √3
= 3√3 + 2√3
• Question 4

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

√27 - √12

√3
EDDIE SAYS
27 = 9 × 3
12 = 4 × 3
9 × 3 - √4 × 3 = √9 x √3 - √4 × √3
= 3√3 - 2√3
• Question 5

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

√80 - √20

2√5
EDDIE SAYS
80 = 16 × 5
20 = 4 × 5
16 × 5 - √4 × 5 = √16 x √5 - √4 × √5
= 4√5 - 2√5
• Question 6

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

√27 + √48

7√3
EDDIE SAYS
27 = 9 × 3
48 = 16 × 3
9 × 3 + √16 × 3 = √9 x √3 + √16 × √3
= 3√3 + 4√3
• Question 7

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

√50 + √98

12√2
EDDIE SAYS
50 = 25 × 2
98 = 49 × 2
25 × 2 + √49 × 2 = √25 x √2 + √49 × √2
= 5√2 + 7√2
• Question 8

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

√63 - √28

√7
EDDIE SAYS
63 = 9 × 7
28 = 4 × 7
9 × 7 - √4 × 7 = √9 x √7 - √4 × √7
= 3√7 - 2√7
• Question 9

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

√300 + √12

12√3
EDDIE SAYS
300 = 100 × 3
12 = 4 × 3
100 × 3 + √4 × 3 = √100 x √3 + √4 × √3
= 10√3 + 2√3
• Question 10

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

√275 - √44

3√11
EDDIE SAYS
275 = 25 × 11
44 = 4 × 11
25 × 11 - √4 × 11 = √25 x √11 - √4 × √11
= 5√11 - 2√11
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