Property 1: Reciprocal Property

A number in exponential notation can also be expressed in terms of its reciprocal. The reciprocal of a number raised to the exponent a is that number raised to the exponent -a .
Therefore,

X ^a = 1 / X^-a and
X ^-a = 1 / X^a

Examples:

2^-3 = 1 / 2³ = 1 / 8

5² = 1 / 5^-2 = 1 / (1 / 25) = 25

When two exponential numbers having same base are multipled, their exponents are added.

If k and t are numbers, then

X^k * X^t = X^k+t Example: 243 * 81 = 3⁵ * 3⁴ = 3⁵⁺⁴ = 3⁹ = 19,683

It is important to note that the two exponential numbers to be multiplied should have the same base.

Example: 16 * 1024 = 2⁴ * 4⁵ = 2⁴ * (2²)⁵ = 2⁴X 2¹⁰ = 2¹⁴ = 16,384

When an exponential number is divided by another exponential number with the same base, their exponents are subtracted.

If k and t are numbers, then

X^k / X^t = X^k-t

Example: 243 / 81 = 3⁵/ 3⁴ = 3^5-4 = 3¹ = 3

It is important to note that the two exponential numbers to be divided should have the same base.

Example: 64 / 16 = 2⁶ / 4² = 2⁶ / (2²)² = 2⁶/ 2⁴ = 2^6-4 = 2² = 4

When two bases with the same exponent are multiplied, their product is equal to the product of the bases with that exponent. X ^a Y^a = (XY)^a

When two bases with the same exponent are divided, their quotient is equal to the quotient of the bases with that exponent. X^a/Y^a = (X/Y)^a