Surds

Question 1

What is a surd?

Answer 1

An irrational root of a rational number

E.g
√5
√40
√27

They are not square numbers

Question 2

Simplifying surds

Answer 2

√3 × √5 is the same as √3×5 = √15

Question 3

How do surds simplify (dividing 1 into two)

Answer 3

You can write them to any of its factor pairs

√40 = √1×√40, √2×√20, √4×√10 or √5×√8

It could also be written as √80÷√2

This does not work with addition or subtraction

Sometimes these factor pairs contain square numbers like √4×√10
This can then be simplified to 2×√10 or 2√10

Question 4

What factor should you choose when simplifiying surds

Answer 4

Always find the biggest square number

Question 5

What happens when you multiply a surd by itself?

Answer 5

If you multiply a surd by itself the root disappears
√4×√4 = 4

Question 6

How can you add or subtract surds

Answer 6

They must have the same numbers inside (like how fractions need the same denominators)

The numbers outside would add if the numbers inside are the same
√1 + √2 is not possible
√3 + √3 would be 2√3
But
2√3 + 4√3 would be 6√3

Question 7

How do you simplify an expression that contains surds?

E.g write √125 - 2√45 +(√5 +2)² in the form of a + b√5 where a and b are integers

Answer 7

You must simplify the expression until it appears in the form wanted

√125 = √5×25 = 5√5

2√45 = 2 × √9 × √5 = 2 × 3 × √5 = 6√5

(√5+2)^2 = (√5 + 2) × (√5 + 2) = 5 + 2√5 + 2√5 + 4 simplifies to 9 + 4√5

Put back into the question as

5√5 - 6√5 + 9+ 4√5

Now simplify this making :

3√5 + 9
So we have 9 + 3√5

Question 8

Simplifying surds further e.g √18

Answer 8

First find the factors of 18

1, 18
2,9

Chose the factor with a square number and replace the √18

√2×9 = √2×√9

As √9 is 3 this becomes √2. × 3 or
3√2