Intermediate Algebra/Exponents

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An exponential function is a shortcut for multiplication, just as multiplication is a shortcut for addition. For example:

When multiplying exponents of the same base, simply add the exponents. For example:

When distributing an exponent, multiply any exponents. For example:

(x³)⁶ = x¹⁸

[edit] Scientific Notation

Scientific notation makes use of exponents. It is often used for very large or very small numbers. It's easier to write 1,574,000,000,000,000 as 1.574 * 10^15. To convert from regular notation to scientific notation, find the leftmost nonzero digit. Count how many places away it is from the ones digit. If the digit was on the right of the ones digit, the exponent will be negative. If it was the ones digit, the exponent will be zero. For everything to the left it will be positive. Then, move the decimal place of the original number so that exactly one nonzero digit is on the left. Write down this new number and * 10^(exponent). You're done!

[edit] Properties of Exponents

1. Product Rule
   
   
2. Quotient Rule
   
   
3. Zero - Exponent Rule
   
   0⁰is undefined
   ( − 4)² = 16
   − 4² = − 16
   
4. Power Rule
   
   Example:
   
5. Negative Exponent Rule
   b ^{− n} =
   
6. Product - to - powers
   
   
   
7. Quotient - to - powers
   

Note: Common mistakes made

Problems: 1. Motivation for the negative exponent rule: Since positive exponents are defined by repeated multiplication, show that if then a^ba^c = a^{b + c} for all b and c.

2. Motivation for the non-definition of 0⁰: Try to define 0⁰. A good definition would ensure that a^ba^c = a^{b + c}. Show that there is no good definition. (You may find it useful that it is impossible to define 1/0 consistently -- why?)

3. The importance of the product rule. Show that b⁰ = 1 and are consequences of the product rule.

4. Fractional powers. What would be a sensible definition of a^{1 / 2}? a^{m / n}?