Language learning | English » Raising to a power, the power function

Raising to a power, the power function Raising to the n-th power if n€Z and n>0 -in this case raising to a power is multiplication with identical terms -let a€R and n>2: an is a progression (sequence) with all its terms = a an = a * a a a ↑ n times - n shows how many times the base (a) should be taken as a term in the multiplication -a1 = a by definition Identities: -multiplication of ‘powers’ with identical bases an * ak = an+k proof: due to definition of raising to a power and associative property of multiplication ↓ an * ak = (a a a) (a a) = an+k ↑ n times ↑ k times ↑ n+k times -division of powers with identical bases a€R ; n,k€Z+ if n>k: an = an-k ak ↑ by def. of raising to a power and rules of simplification if n = k if n < k an = 1 ak an = 1 ak ak-n -raising a power to a power n,k€Z+ (an)k = ank -raising a product to a power (ab)n = (ab)(ab)(ab) = (a * a a)(b b b) = an bn ↑ by definition

Commutative: Associative: ↑ commutative and assoc. characteristics of multiplication ↑ by definition A*B = BA (A * B) C = A (B C) -raising a fraction to a power (b is not 0) (a/b)n = an / bn Identities with 0, negative and irrational powers: ‘Carried out’ in a way that all the previous features/identities of the process remain the same. (permanenciaelv) -raising to the power of 0 a≠0 a0 = 1 -raising to the power of –(n), where nEZ+ a-n = 1 an -power with rational exponents ap/q = q√(ap) If the base is 0, the result with positive exponents is 0. With negative exponents, there’s no sol. Power – functions xx3 if n is odd and nEZ : xxn domain: R range: R not bounded strictly monotonously increasing no extrema odd x;y intersection at the origin if n is even and nEZ xxn domain: R range: [0;∞[ ‘alulról korlátos’ at 0 ] -∞; 0] str. mon decreasing, [0; ∞[ str mon increasing minimum at (0;0) even x; y intersection at the origin Applications:

Mathematics: Probability calculations Surface area, Volume calculations Physics: Calculations with radioactive decay xx2