## Exponents

### Lesson No. 2:

Solved Problems in Exponents:

Example 1:

Simplify:

a. (81)3/4
Solution:
(81)3/4 = (34)3/4 = (3)4 × 3/4 = (3)4 × 3/4 = 33 = 27

b.(64/25)-3/2
Solution:
(64/25)-3/2 = (82/52)-3/2
=(8/5)2-3/2 = (8/5)2 ×-3/2 = (8/5)2 ×-3/2
(8/5)-3 = (5/8)3 = 53/83 = 125/512

c. If 10p = 0.000001, then find p.
Solution:

0.000001 = 1/1000000 = 1/106 = 10-6
Now, 10p = 10-6
So we get p = -6

Example 2:

Express 23 + 24 as a product of two prime factors.
Solution:

First of all note that:
23 + 24 ‡ 27
Now, in 23 and 24 , 23 is a common factor
23 + 24 = 23 (1+2) = 23 × 3

Example 3:

Find the greatest prime factor of:
23 + 24 +25 + 26 + 27
Solution:

23 + 24 +25 + 26 + 27 = 23 (1 + 2 + 22 + 23 + 24) = 23 × 31
Since 31 is a prime number, and the sum
23 + 24 +25 + 26 + 27 is same as 23 × 31.
Therefore, 31 is the greatest prime factor in
23 + 24 +25 + 26 + 27

Example 4:

Simplify
26 × 27 - 24 × 29

Solution:

26 × 27 = 26 + 7 = 213 and 24 × 29 = 24 + 9 = 213
Therefore, 26 × 27 - 24 × 29 = 213 - 213 = 0

Example 5:

Find 23 + 2-3 + 20
Solution:

We know that 2-3 = 1/23 = 1/8 and 20 = 1
So, 23 + 2-3 + 20 = 8 + 1/8 + 1 = 9 + 1/8
LCM is 8, and therefore 9 + 1/8 = (9 × 8 + 1)/8 = (72 + 1)/8 = 73/8

Example 6:

Let a and b be positive integers such that
ab = 121, then find ba .

Solution:
Since a and b are positive integers and
ab = 121, so, we can write 112 =121
Therefore, a = 11 and b = 2.
Now, ba = 211 = 2048

Example 7:

Compare
a. -23 and -24.
b. -1/23 and -1/24

Solution:

-23 = -8 and -24 = -16
Now, -16 < -8
Therefore, -23 > -24

Very Important Tip:
(-2)4 = -2 × -2 × -2 × -2 = 16, but -24 = -16

The negative sign of the base will be gone, only when the base is inside a parentheses and the exponent is an even integer.

b. -1/23 = -1/8, and -1/24 = -1/16
Now, we know that:
8 < 16, -8 > -16 and again -1/8 < -1/16

Very Important Tip:

The inequality sign reverses direction in two cases:

Case 1:

When negative sign is either introduced or pulled out.

For Example:
2 < 3, but -2 > -3 Or
-10 > -11, but 10 < 11

Case 2:

When reciprocals are taken or removed:
The reciprocal of any number x, except 0 is 1/x
The reciprocal of 3 is 1/3

Example:
10 < 14 but 1/10 > 1/14 Or
1/2 > 1/3, but 2 < 3, and again
2 < 3, but -2 > -3, and again -1/2 < -1/3

Example 8:

Which of the two is larger?
3√5 or 4√7

Solution:

3√5 is a surd of order 4 and 4√7 is a surd of order 3.

To compare two surds:

First find LCM of the two orders: i.e. 3 and 4.
LCM of 3 and 4 is 3 × 4 = 12
Now, 3√5 = 51/3 = (54/3×4) = 54/12 = (625)1/12 = 6251/12
Again 4√7 = 71/4 = (73/4×3) = 73/12 = (73)1/12 = 3431/12
Now, 6251/12 > 3431/12