Surd
A surd is simply a number that has a square root (or cube root, etc.) that can't be simplified to a whole number. In other words, it's an "irrational" root.
For example:
√9
√16
√2
√3
√7
Think of it this way: if you try to calculate √2 on your calculator, you get 1.4142135... and the decimals go on forever without any pattern. That's why we keep it as √2 instead of writing the decimal.
Some important things to remember about surds
We can simplify some expressions containing surds. For example:
√8 = √(4 × 2) = √4 × √2 = 2√2
We can add or subtract surds only if they're "like terms":
3√2 + 2√2 = 5√2
But you can't simplify 2√2 + √3 any further.
When multiplying surds:
√2 × √3 = √6
√2 × √2 =
Do you remember how to simplify the following?
(i)
(ii)
(iii)
(iv)
To get these answers, you would have used the following laws of exponents, which you have learnt in your earlier classes (Here a, n and m are natural numbers.
Remember a is called the base and m and n are the exponents.)
| (i) | |
| (ii) | |
| (iii) | |
| (iv) |
What is
it is
So you have learnt that
So, using (iii), we can get
We can now extend the laws to negative exponents too.
For example:
(i)
(ii)
(iii)
(iv)
Example 20: Simplify
How would we go about it?
It turns out that we can extend the laws of exponents that we have studied earlier, even when the base is a positive real number and the exponents are rational numbers.
We define for a real number a > 0 as follows:
Let a > 0 be a real number and n a positive integer. Then
In the language of exponents, we define
There are now two ways to look at
Therefore, we have the following definition:
Let a > 0 be a real number. Let m and n be integers such that m and n have no common factors other than 1, and n > 0. Then,
We now have the following extended laws of exponents: Let a > 0 be a real number and p and q be rational numbers.
Then, we have
| (i) | (ii) |
| (iii) | (iv) |