Important formulae :

a) Indices :

* an = a . a . a ......(n times)

* am . an . ap = am + n + ..... + p

* am . an = am + n

* straight a to the power of straight m over straight a to the power of straight n space equals space straight a to the power of straight m space – space straight n end exponent

* (am)n = amn = (an)m

* (ab)n = anbn

* open parentheses straight a over straight b close parentheses to the power of straight n space equals space straight a to the power of straight n over straight b to the power of straight n

* straight a to the power of straight m to the power of straight n end exponent space equals space open parentheses straight a to the power of straight m close parentheses to the power of straight n space equals space straight a to the power of left parenthesis straight m to the power of straight n right parenthesis end exponent

* straight a to the power of – straight n end exponent space equals space 1 over straight a to the power of straight n

* straight a to the power of straight n space equals space 1 over straight a to the power of – straight n end exponent

* straight a to the power of 0 space equals space 1 where a ∈ r, a ≠ 0

* straight a to the power of straight p over straight q end exponent space equals space straight q-th root of straight a to the power of straight p end root

* If am = an Where a ≠ 0, a ≠ ± 1 then m = n

* an = bn Where n ≠ 0 then a = ± b

b) Surds :

Let 'a' be a rational number and 'n' be a positive integer such that straight n-th root of straight a is irrational. then straight n-th root of straight a is called surd of order 'n'.

i.e. → open parentheses 5 close parentheses 1 over 7 space equals space root index 7 of 5 is a surd of order 7.

* Every surd is a irrational number.

Rules of surds :

* square root of straight m space cross times space square root of straight n space equals space square root of mn

* fraction numerator square root of straight m over denominator square root of straight n end fraction space equals space square root of straight m over straight n end root

* square root of straight a. space square root of straight a. space square root of straight a space.... space straight n end root end root end root times = straight a to the power of 1 space – space 1 over 2 to the power of straight n end exponent

* square root of straight a square root of straight a square root of straight a square root of straight a... infinity end root end root end root end root space equals space straight a

* square root of straight a space plus space square root of straight a space plus space square root of straight a space plus space.... infinity end root end root end root space equals space straight p then p(p – 1) = a

* Conjugate surd : square root of straight a space plus space square root of straight b space equals space plus-or-minus space open parentheses square root of straight a space – space square root of straight b close parentheses

* Condition of equality of surds

If straight a space plus space square root of straight b space equals space straight c space plus space square root of straight d

then a = c, b = d

To arrange the surds in increasing or decreasing order →

If the given surds are straight a to the power of 1 over straight m end exponent comma space straight b to the power of 1 over straight n end exponent comma space straight c to the power of 1 over straight p end exponent so the steps to be followed for comparison.

(i) First find the LCM of  (m, n, p) order of surds.

(ii) Then the numerator and denominator of power is multiplied by such a number so that denominator of all terms becomes equal to the LCM.

Example :

Compare fourth root of 3 comma space cube root of 2 comma space root index 6 of 5

left parenthesis 3 right parenthesis to the power of 1 fourth end exponent comma space left parenthesis 2 right parenthesis to the power of 1 third end exponent comma space left parenthesis 5 right parenthesis to the power of 1 over 6 end exponent

LCM of (4, 3, 6) = 12

So, left parenthesis 3 right parenthesis to the power of 1 fourth space cross times space 12 end exponent comma space left parenthesis 2 right parenthesis to the power of 1 third space cross times space 12 end exponent comma space left parenthesis 5 right parenthesis to the power of 1 over 6 space cross times space 12 end exponent

rightwards double arrow space left parenthesis 3 right parenthesis cubed comma space left parenthesis 2 right parenthesis to the power of 4 comma space left parenthesis 5 right parenthesis squared

= 27, 16, 25

So, order is ⇒ fourth root of 3 space greater than space root index 6 of 5 space greater than space cube root of 2

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