27213 272/3 v27) 9 and U and 9 (J) The arithmetic properties of exponents are us

Question

27213 272/3 v27) 9 and U and 9 (J) The arithmetic properties of exponents are used to simplify expressions involving general exponents. Arithmetic Properties of Exponents ri and rz, we have For a positive a and numbers ari ari rir2 (ari) and ritrz a' a arz For a pair of positive real numbers an and G d a single rational number r, we have (aiaz)" and Exponents can be extended to include all real numbers, not just rational numben We will define, for every positive number a 1, an exponential function f(r) that is valid for all real numbers x. The definition must ensure that the arithmet properties of exponents continue to hold, and the definition must reduce to the and power definition when the exponent is rational.

Explanation / Answer

explaining the properties by an example

4^5 4^ 2

when base of exponent is same and we multiply then we add the exponents

4^5 4^2 = 4^ (5+2) = 4^7

( 4^5 / 4^2)

when we divide same base with different exponents then exponents are subtracted

( 4^5 / 4^2) = 4^(5-2) = 4^3

( 4^5 )^2

the outside exponent is multiplied with exponent inside

(4^5 )^2 = 4^10

----> 4^5 * 2^ 5

when base is different and exponents are same

we can multiply the base and put the exponent on the result

(4*2)^5 = 8^5

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