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Summary | R
From the beginning the base of the exponent could be any known number. However the exponent was a natural number greater than 1. Thereafter 1, fractions, 0, and negative numbers were allowed to be exponents in order.

1 Repeated Multiplication | R
Multiplication is repeated addition. a × 3 = a + a + a Exponents are basically repeated multiplication. a ^ 3 = a × a × a

2 Inverse Operator, Root | R
Addition and subtraction are inverse operations. a + b = c; c - b = a; Multiplication and division are inverse operations. a × b = c; c ÷ b = a; Exponents and roots are inverse operations. a ^ b = c; The b-th root of c is a.

3 Irrational Number | R
According to Pythagoras' theorem the diagonal length of a square with each side measuring one unit is the square root of 2. The assumption that square root of 2 could be expressed as a ratio of two integers deduces a contradiction.

4 The power of 1 | R
a^3 is a×a×a and a^2 is a×a, so define a^1 as a.

5 The power of a fraction | R
(a^b)^c = a^(b×c) and a^1 = a, so if we define a^(1/b) as the b-th root of a, then (a^b)^(1/b) = a^(b×(1/b)) = a^1 = a

6 The power of 0 | R
The limit of a^x as x approaches 0 from the right is 1. So define a^0 as 1.

7 The power of a negative number | R
(a^b) × (a^c) = a^(b+c) and a^0 = 1, so if we define a^(-b) as 1/(a^b), then (a^b) × (a^(-b)) = a^(b+(-b)) = a^0 = 1

8 Inverse Operator, Logarithm | R
a ^ b = c An inverse operator only when b is a natural number greater than 2: The b-th root of c is a. Another inverse operator when b is a real number: The log base a of c is b.

Reading b^n | R
"b raised to the power of n", "b to the n-th power", "b to the n-th", "b to the n", "the n-th power of b", "b squared", "b cubed"

Saying Logarithm | R
"The base of the log raised to the power of what's on the other side of the equal sign will equal the number that the log is operating on."

Saying Logarithm | R
If you want to find log base b of x, you're asking "What power do I have to raise b to in oder to get x?" Or you can say, "If I raise b to the power of y, I'm going to get x."

Triangle of Power | R
Triangle of Power
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