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In this article, we will be looking at rules of indices and surds.

Laws of indices

1. am × an = a(m + n)

e.g. 23 × 24 = 2(3 + 4)  = 27

2. am ÷ an = a(m – n)

e.g. 28 ÷ 25 = a(8 – 5) = 2

3. (am)n = amn

e.g. (22)3 = 26

4. a0 = 1

5. a-1 = 1/a

6. a-m = 1/am

7. a1/2 = √a

8. a1/m = m√a

9. an/m = (m√a)n

e.g. 25/3 = (3√2)5

Surds

The term surd traces back to al-Khwārizmī (c. 825), who referred to rational and irrational numbers as audible and inaudible, respectively. This later led to the Arabic word for irrational number being translated into Latin as “surdus” (meaning “deaf” or “mute”).  An unresolved root, especially one using the radical symbol, is often referred to as a surd. In other words, surds are irrational numbers. For expressions involving surds, rationalizing the denominator is the most commonly used method of simplification. This is used for expressions where a surd occurs in the denominator.

2/√3

As √3 can not be simplified further and it is present in the denominator, multiplying both the numerator and the denominator will remove the root sign from the denominator.

2*√3 / √3*√3

=2√3 / 3

√(338/16)

After canceling out 2 from numerator and denominator, we are left with √(169/8) = 13/2√2

To simplify this further, multiply both numerator and denominator with √2 which will give us 13√2/4.

3 / (2 + √2)

In the denominator, we have 2 + √2. If this is multiplied by 2 – √2; it will be in the format (a + b)(a – b) = a– b2

To rationalize the denominator we will have to multiply by 2 – √2

3 (2 – √2) / (2 + √2)(2 – √2)
6 – 3√2 / (4 – 2)
6 – 3√2 / 2
So, if the denominator is in the form (a + b), multiply the numerator and the denominator with (a – b). If the denominator is in the form (a – b), multiply the numerator and the denominator with (a + b).
Indices and surds have applications in other question types esp. in number system and algebra based questions.
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