State with reason, which of the following are
surds.
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Answered by
1
Answer:
, or 1
1
2
. So some square roots can be evaluated as whole numbers or
as fractions, in other words as rational numbers. But what about √
2 or √
3? The roots to these
are not whole numbers or fractions, and so they have irrational values. They are usually written
as decimals to a given approximation. For example
√
2 = 1.414 to 3 decimal places,
√
3 = 1.732 to 3 decimal places.
When we have square roots which give irrational numbers we call them surds. So √
2 and √
3
are surds. Other surds are
√
5,
√
6,
√
7,
√
8,
√
10 and so on.
Surds are often found when using Pythagoras’ Theorem, and in trigonometry. So, where possible,
it is useful to be able to simplify expressions involving surds. Take, for example, √
8. This can
be written as √
4 × 2, which we can rewrite as √
4 ×
√
2, in other words as 2
√
2:
√
8 = √
4 × 2
=
√
4 ×
√
2
= 2√
2 .
In general, the square root of a product is the product of the square roots, and vice versa. This
is useful to know when simplifying surd expressions.
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Now suppose we have been given √
5 ×
√
15. At first glance this cannot be simplified. But we
can rewrite the expression as the square root of 5 times 15, so it is the square root of 75, and
75 can be written as 25 times 3. But 25 is a perfect square, we can use this to simplify the
expression.
√
5 ×
√
15 = √
5 × 15
=
√
75
=
√
25 × 3
=
√
25 ×
√
3
= 5√
3 .
But watch out if you are given √
√
4 + 9, which is the square root of 13. This does not equal
4 + √
9, which is 2 + 3 = 5. Now 5 cannot be the answer as that is the square root 25, not
the square root of 13.
Key Point
If a positive whole number is not a perfect square, then its square root is called a surd. A surd
cannot be written as a fraction, and is an example of an irrational number.
4. Simplifying expressions involving surds
Knowing the common square numbers like 4, 9 16, 25, 36 and so on up to 100 is very helpful
when simplifying surd expressions, because you know their square roots straight away, and you
can use them to simplify more complicated expressions. Suppose we were asked to simplify the
expression √
400 ×
√
90:
√
400 ×
√
90 = √
4 × 100 ×
√
9 × 10
=
√
4 ×
√
100 ×
√
9 ×
√
10
= 2 × 10 × 3 ×
√
10
= 60√
10 ,
which cannot be simplified any further.
We can also simplify the expression √
2000/
√
50. We get
√
2000
√
50
=
r
2000
50
=
√
40
=
√
4 × 10
=
√
4 ×
√
10
= 2√
10 .
www.mathcentre.ac.uk 5 c mathcentre 2009
Key Point
The following formulæ may be used to simplify expressions involving surds:
√
ab =
√
a ×
√
√
3) = 1 − √ 3 + √ 3 − √ 3 √ 3 = 1 −
Step-by-step explanation:
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