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Solution :

(i) `(12^4*9^3*4)/(6^3*8^2*27)` <br>
`= ((2^2*3)^4*(3^2)^3 *2^2)/((2*3)^3 *(2^3)^2 * 3^3)` <br>
`= (2^8*3^4*3^6*2^2)/(2^3*3^3*2^6*3^3)` <br>
`= 2^(8+2-3-6)*3^(4+6-3-3)` <br>
`= 2*3^4 = 162` <br>
(ii) `2^3*a^3*5a^4` <br>
`= 2^3*a^3*a^4*5` <br>
`= 2^3*5*a^(3+4)` <br>
`= 8*5*a^7 = 40a^7` <br>
answer**Introduction**

**In general if n is a natural `a xx a xx a xx a xx ...xx a = a^n `n - times a^n is called the nth power of a and is a also read as a raised to the n**

**In is evident from the above discussion that `a xx a xx a xx b xx b ` is written as `a^3 b^2` (read as a cubed into b squared ) ` a xx a xx b xx b xx b xx b` is written `a^2 b^4 ` (read as a squared into b raised to the power4)**

**For any non - zero rational number a we define `(i) a^1 = a` (ii) `a^@=1` .**

**If following form the above example that `(-1)` an odd natural number `=-1 and (-1)` An even natural number `=1` i.e. `(-1)^n={1` if n is an even natural number and `-1` if n is odd natural number**

**If `a/b` is a rational number and n is a natural number then `(a/b)^n=a/b xx a/b xx a/b xx a/b xx.....xx a/b = (a xx a xx a xx a xx....xx a)/(b xx b xx b xx b xx.......xx b)=a^n/b^n`**

**Express the following numbers as product of powers of their prime factors : (i) `1000` (ii) `16000` (iii) `3600`**

**Mul tipl ying powers with the same best**

**First law if is any non -zero rational number and` m n ` are natural numbers then `a^m xx a^n = a^(m+n) `**

**Generalisation : if a is a non - zero rational number and `m,n,p ` are natural numbers then `a^m xx a^n xx a^p = a^(m+n+p)`**