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 10^p = 0.000001, then find p.
Solution:
0.000001 = 1/1000000 = 1/10⁶ = 10^-6
Now, 10^p = 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 = 2^{6 + 7} = 2¹³ and 2⁴ × 2⁹ = 2^{4 + 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
a^b = 121, then find b^a .
Solution:
Since a and b are positive integers and
a^b = 121, so, we can write 11² =121
Therefore, a = 11 and b = 2.
Now, b^a = 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 = 5^1/3 = (5^4/3×4) = 5^4/12 = (625)^1/12 = 625^1/12
Again ⁴√7 = 7^1/4 = (7^3/4×3) = 7^3/12 = (7³)^1/12 = 343^1/12
Now, 625^1/12 > 343^1/12