Exponents

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. 6³ could be referred to as 6 to the 3rd power or simply 6 cubed.


Zero Power & First 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)

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

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 = - 64 

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 x 1/5⁴ x 1/5⁴ x 1/5⁴ = 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 = (34)/(24) = 81/16 

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 

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

Power Rule (Raising exponential expressions to a power)

To raise an exponential expression to a power, multiply the expressions.

(3²)³ = 3^2x3 = 3⁶

Fractional Exponents

Laws (Rules) of Exponents

Practice

Khan:  Exponents intro and order of operations unit

Reference
Math.com: Exponents
Wikipedia: Exponentiation
About.com: What are Exponents? An Overview of Exponents
Sparknotes: Powers, Exponents, and Roots
Cuemath: Exponents

Prodigy: Exponent Rules: 7 Laws of Exponents to Solve Tough Equations

https://www.mathplanet.com/education/pre-algebra/discover-fractions-and-factors/powers-and-exponents

https://www.mathsisfun.com/algebra/exponent-fractional.html

https://www.cuemath.com/algebra/fractional-exponents/