 # Simplifying Surds (5)

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√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

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

 2√5 - 6 x 5 3 x 5

which simplifies to

 2√5 - 30 15

which equals

 2√5 -   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)

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)

-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)

-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)

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

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

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

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

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)

-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

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

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