* 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 (ab), a is called base, b is called index (or) power (or) exponent.
* If the given bases are same then, their powers are equal.
Ex: xa = xb then a = b
* If the given powers are same then, their bases are equal.
Ex: an = bn then a = b
Laws of Indices
1) am × an = am + n
2) am ÷ an = am - n
3) (am)n = amn
5)
6) a0 = 1
Important formulae
*
* am = bm a = b
* am = an m = n
*
*
1. (27)8.2 × 3x = 2710
Ans: 5.4
Exp: 33 × 8.2 × 3x = 33 ×10
24.6 + x = 30 x = 5.4
Ans: 2
Exp:
Bases are equal Powers are equal.
x − 1 = 3 − x x = 2
3. xa(b − c) . xb(c − a) . xc(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 2x = 3y = 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 100.48 = x, 100.70 = y and xz = y2 then, z is approximately equal to .......
Ans: 2.9
Exp:
xz = y2 10(0.48z) = 10(2 × 0.70)
1.40 0.48 z = 1.40
= 2.9 (approx)
17. Find the value of (xb + c)b − c (xc + a)c − a (xa + 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.