Math & History: July 2016

I. Intro

Exponents are a shorthand way to show how many times a number, called the base, is to be multiplied times itself. A number with an exponent is said to be "raised to the power" of that exponent.

Hence with the above example, we would say four is being raised to the second power, which means 4 x 4.

Some examples of exponents are:

3 × 3 × 3 × 3 × 3 = 3⁵
-2 × -2 × -2 = (-2)³

Squares and Cubes
Common exponents have their own names. The exponent 2 is referred to as "squared" and the exponent 3 is referred to as "cubed." For instance, 5² in words could be called 5 to the second power or 5 squared. 6³could be referred to as 6 to the 3rd power or simply 6 cubed.

II. Laws (Rules) of Exponents

1. Zero Power & First Power

Zero Power
A number (except zero) raised to the zero power equals 1. Example 4⁰= 1.

*Khan: The Zero Power (the 2nd explanation makes sense to me)

First Power

A number (except zero) raised to the 1st power equals the number itself. Example, 4¹= 4.

2. Negative Base with Positive Exponent
A negative number (negative base) with a positive exponent is simply the negative number multiplied by itself however number of times as indicated by the exponent. Following the rules for multiplying integers this means that:

If a negative number (base) is raised to a positive even power, the result is a positive number

(- 5)2 = - 5 ×- 5 = 25

If a negative number (base) is raised to a positive odd power, the result is a negative number

(- 4)3 = - 4 × -4 × - 4 = 16 x -4 = - 64

3. Negative Exponents
A negative exponents means that we take the inverse of the base and multiple it by itself the number of times indicated by the exponent.

5^-4= 1/5⁴ = 1/5 x 1/5 x 1/5 x 1/5 = 1/625

If the base number is a fraction, then the negative exponent switches the numerator and the denominator.

(2/3)-4 = (3/2)4 = (3/2) x (3/2) x (3/2) x (3/2) = 81/16

4. Product Rule (Multiplying exponential expressions with the same base)

When multiplying two bases of the same value, keep the bases the same and then add the exponents together to get the solution

3² x 3⁴ = 3 ^{2 + 4} = 3⁶

5. Quotient Rule (Dividing exponential expressions with the same base)

To divide exponents with the same base, subtract the exponents.

3⁴ ÷ 3² = 3^4-2 = 3²

6. Power Rule (Raising exponential expressions to a power)

To raise an exponential expression to a power (power is raised to another power), multiply the expressions.

(3²)³ = 3^2x3 = 3⁶

7. Power of a Product Rule

The power of a product rule states that when you raise a product of two or more numbers to a power, you can distribute the exponent to each factor

(2x)3 = 23 ⋅ x3 = 8x3

8. Power of a Quotient Rule (Power of a Fraction Rule)
The Power of a Quotient Rule states that when a fraction (quotient) is raised to a power, you distribute the outer exponent to both the numerator and the denominator.

(2x/3y)³ = (2x)³ / (3y)³ = 8x³ / 27y³

9. Fractional Exponents