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Exponents by Mind Map: Exponents

1. Degree

1.1. Polynomials

1.1.1. Number of roots

1.1.1.1. Real

1.1.1.2. Complex

2. Repeated Multiplication

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

3. Logarithms

3.1. Inverse of Exponents

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

4. Laws

4.1. Multiplication

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

4.2. Division

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

4.3. Power of a Power

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

4.4. Power of a Product

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

4.5. Power of a Quotient

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

5. Integer Exponents

5.1. Alternate Notations

5.1.1. Scientific Notation

5.1.1.1. Must have exactly one significant digit before the decimal

5.1.2. Engineering Notation

5.1.2.1. exponents for the powers of ten are multiples of 3 to coincide with the metric prefixes

5.2. positive

5.3. negative

5.4. 0

5.4.1. a^0 = 1

6. Rational Exponents

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

7. Complex Exponents

7.1. Trigonometric Functions

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

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

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

7.2. e^(x+iy)

7.2.1. real portion e^x is magnitude

7.2.2. imaginary portion e^iy is direction

8. Repeated Division

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

9. Exponential Function

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

9.1.1. b < |1|, reflection in y-axis

9.1.2. a < 0, reflection in x-axis

9.1.3. if a > 0, b > 1

9.1.3.1. Domain : x e R

9.1.3.2. Range : y > 0 , y e R

9.1.3.3. No x-intercepts

9.1.3.4. y intercept at (0,1)

9.1.3.5. Asymptote @ y = 0

9.2. Exponential Decay

9.2.1. Half-Life

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

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

9.3. Exponential Growth

9.3.1. Compound Interest

9.3.2. Moore's Law for transistors

9.3.3. Nuclear Reaction

10. Origins

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

10.2. 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"

11. Real Exponents

12. Composition of Functions

12.1. Exponentiation as repeated composition of functions

13. Special Bases

13.1. 10

13.1.1. Standard Place Value; Decimal Number system

13.2. 2

13.2.1. Binary Number System

13.3. 16

13.3.1. Hexadecimal Number System

13.4. 8

13.4.1. Octal Number System