THEORY OF INDICES

* In an exponential form, convert the given multiplication or division in such a way that either the bases are same or the powers are same.
* Check for possibility of applying Laws of Indices.
In an exponential form (a^b), a is called base, b is called index (or) power (or) exponent.
*  If the given bases are same then, their powers are equal.
Ex: x^a = x^b then a = b
*  If the given powers are same then, their bases are equal.
Ex: aⁿ = bⁿ then a = b
 

Laws of Indices

1) a^m × aⁿ = a^{m + n}
2) a^m ÷ aⁿ = a^{m - n}
3) (am)ⁿ = a^mn

5) 
6) a⁰ = 1

Important formulae

* 
* a^m = b^m   a = b
* a^m = aⁿ   m = n
* 
* 

1. (27)^8.2 × 3^x = 27¹⁰
Ans: 5.4
Exp: 3^{3 × 8.2} × 3^x = 3^{3 ×10}
24.6 + x = 30   x = 5.4

Ans: 2
Exp: 

Bases are equal  

 Powers are equal.
x − 1 = 3 − x   x = 2

3. x^{a(b − c)} . x^{b(c − a)} . x^{c(a − b)} = ?
Ans: 1
Exp: x^{(ab − ac)}. x^{(bc − ba)} . x^{(ca − cb)}
= x^{(ab − ac + bc − ba + ca − cb)}
= x⁰ = 1

Ans: 1
Exp: 

Ans: a
Exp: 

6. 

 
Ans: 

Exp: i) 33³³ = (3³ to 3⁴)³³ = 3⁹⁹ to 3¹³²
                                   (    33 is in between 3³ to 3⁴)
ii) 33³³ = (3⁵ to 3⁶)³ = 3¹⁵ to 3¹⁸
                                   (    333 is in between 3⁵ to 3⁶)
iii)   = 
iv) 3³³³
As all the numbers are powers of 3, compare (99 − 132), (15 − 18), 3²⁷, 333.
3²⁷ is the highest number.

 is the highest.
Shortcut:
Ignore the same bases (here 3), then consider the powers. The highest no. in the powers will finally be the greatest number.

7. If 2^x = 3^y = 6^−z then, 
Ans: 0
Exp:
     2^(x) = 3^(y) = 6^(−z) = K

8. If a + b + c = 0 then, 
Ans: x³
Exp: 

    [     a + b + c = 0  a³ + b³ + c³ = 3abc]
Shortcut:
Given condition a + b + c = 0. Consider the values of a, b and c such that the condition is satisfied i.e., (a, b, c) can be (2, −1, −1) or (0, −1, +1), etc.....
Let us take a = b = c = 0

    
     = x⁰ + 0 + 0 = 1

9. If x = 0.9 then, find the value of  

Ans: 0.9
Exp: 

10. What is the value of  
Ans: 2ⁿ
Exp:
Taking 2ⁿ common in the numerator.

11.  when m = 19, n = 17
Exp:

12. Find the values of a and b if,     
Ans: a = 4, b = 1
Exp:
x^(2a) = x^(8b)   a = 4b
x  

 = x   a = 6 − 2b
4b = 6 − 2b   b = 1, a = 4

13. If  

 Find the relation between a and b.
Ans: 

14.  
Ans: 

Exp: 

15.  
Ans: 1
Exp:

16. Given 10^0.48 = x, 10^0.70 = y and x^z = y² then, z is approximately equal to .......
Ans: 2.9
Exp:
x^z = y²   10^(0.48z) = 10^{(2 × 0.70)}
1.40   0.48 z = 1.40 

  = 2.9 (approx)
 

17.  Find the value of (x^{b + c})^{b − c} (x^{c + a})^{c − a} (x^{a + b})^{a − b}
Ans: 1
Exp: 

 x⁰ = 1
(or) Let a = b = c = 0
x⁰ . x⁰ . x⁰ = 1
 

18. If  then, x equal to .....
Ans:  
Exp:

19. If  then, x is equal to ......
Ans: 4
Exp:
5² + 12² = 13²
                   (from the pythogorous triplet)
Hence,  = 2   x = 4
 

20. a³b = abc = 180. a, b, c are positive integers then, the value of c is ......
Ans: 1
Exp:
a³.b = a . b . c  

 a² = c
                        (where a, c are integers).
To satisfy this condition, a, c values must be 1.