Negative Exponents

Exponent is any no. raised to the base which can also be seen as the power of the number that is how many times the number is multiplied by itself. It is represented in the form a^b where a is the base and b is the power. For Example, the number 2⁵ = 2 ∗ 2 ∗ 2 ∗ 2 ∗ 2 i.e 2 multiplied 5 times to itself. Hence here Base = 2, Exponent = 5. This can be read as 2 raised to the power 5. 

Note 1: When there is no given exponent to the number, its power of the number is one i.e. it is one time of itself.

Example: The no. 5  has base 5 has power one +1.

Note 2: Powers are useful for expressing large quantities.

Example: 56,400,000,000,000,000,000 can be expressed easily as 5.64 x 10¹⁸ 

Powers with Negative Exponents

If a number says, n has negative exponent b as its power then it is basically the reciprocal of power. i.e. 

n^-b = 1/n^b

Example: 

The number 2^-3  has base 2 and a negative exponent 3  i.e. -3.  

2^-3 = 1/2³ = 1/(2 * 2*  2) = 1/8

Expanding Rational Numbers with Exponents

A rational number can be expanded and represented in terms of power. 

Example:

45679.32 can be represented as 40000 + 5000 + 600 + 70 + 9 + 0.3 + 0.02 = 4×10⁴ + 5×10³ + 6×10² + 7×10¹ + 9×10⁰ + 3×10^-1 +2×10^-2

Laws of Exponents and Powers

1. Exponents with the same base

• When two exponential numbers are multiplied with the same base, then the exponents are added: n^a * n^b = n^{(a + b)}
• When two exponential numbers are divided with the same base, then the exponents are subtracted: n^a / n^b = n^{(a – b)} 

Example:

(2³ * 2² )/2⁴ = 2^{(3 + 2)}/2⁴  = 2^{5 – 4} = 2

2. Power of Power

When a base having a power of power say (n^a)^b then the powers are multiplied. i.e. n^a*b.

Example:

(3²)³ = 3^2*3 = 3⁶ 

3. Same Exponents but Different Bases

• When two numbers with multiplied with the same exponents then their bases are multiplied. n^a x m^a  = (n*m)^a  
• When two numbers with divided with the same exponents then their bases are divided. n^a / m^a = (n/m)^a

Comparison of Quantities using Exponents

When we need to compare two large or small quantities, we convert them to their standard exponential form and divide them.

Example: Comparing the masses of the earth and that of the sun?

Solution:

Mass of the Earth = 5.972 x 10²⁴ 

Mass of the Sun = 1.989 x 10³⁰

Mass of the Sun/Mass of the Earth = (1.989 x 10³⁰) / (5.972 x 10²⁴)

                                                      = 0.33 x 10⁶ = 3.3 x 10⁵

So the mass of the sun is approximately 10⁵ times that of earth.

Some More Examples

Question 1: Find values of 

(i) 7³
(ii) 7^-3 

Solution:

(i) 7³ = 7 * 7 * 7 = 49 * 7 = 343
(ii) 7^-3 = 1/7³ = 1/(7 * 7 * 7) =1/343

Question 2: Express the following in exponent and powers?
(i) 34500
(ii) 1/25

Solution:

(i) 34500 = 345 x 100 = 345 x 10² 
= 3.45 x 10² x 10²(dividing 345 by 100 by shifting two decimal places to left and at the same 
                                             time multiplying by 100 or 10²)
                = 3.45 x 10⁴         (total power = 2 + 2)
(ii)1/25 = 1/(5 x 5) = 1/5² = 5^-2 (negative exponent)

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