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Exponents by Mind Map: Exponents
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Exponents

Degree

Polynomials

Number of roots, Real, Complex

Repeated Multiplication

a^n = a x a x a x ... x a - n times, where n is a natural number

Logarithms

Inverse of Exponents

y = logb(x) <=> b^y = x

Laws

Multiplication

a^x * a^y = a^(x+y)

Division

a^x / a^y = a^(x-y)

Power of a Power

(a^x)^y = a^(x*y)

Power of a Product

(a*b)^y=a^y*b^y

Power of a Quotient

(a/b)^y = (a^y)/(b^y)

Integer Exponents

Alternate Notations

Scientific Notation, Must have exactly one significant digit before the decimal

Engineering Notation, exponents for the powers of ten are multiples of 3 to coincide with the metric prefixes

positive

negative

0

a^0 = 1

Rational Exponents

a^(1/n) = nth root of a

Complex Exponents

Trigonometric Functions

sin(z)=(e^i*z - e^-i*z)/2i

cos(z)=)e^i*z + e^-i*z)/2

e^(x+i*y)=e^x * (cos(y) + i*sin(y))

e^(x+iy)

real portion e^x is magnitude

imaginary portion e^iy is direction

Repeated Division

a^-n = 1/(a x a x a x ... x a) - n times, where n is a natural number

Exponential Function

F(x) = ab^x, a: coefficient, b: base

b < |1|, reflection in y-axis

a < 0, reflection in x-axis

if a > 0, b > 1, Domain : x e R, Range : y > 0 , y e R, No x-intercepts, y intercept at (0,1), Asymptote @ y = 0

Exponential Decay

Half-Life

Draining of electric charge from a capacitor over time with a constant load

Intensity of EM radiation inside an absorbing medium as a function of distance from the surface

Exponential Growth

Compound Interest

Moore's Law for transistors

Nuclear Reaction

Origins

From the latin "exo-" meaning "out of and "ponere" meaning place.

First introduced by Michael Stifel in 1544 in "Arithmetica Integra" mentioning that "3 is the exponent of 8 and 5 is the exponent of 32"

Real Exponents

Composition of Functions

Exponentiation as repeated composition of functions

Special Bases

10

Standard Place Value; Decimal Number system

2

Binary Number System

16

Hexadecimal Number System

8

Octal Number System