Algebra/Exponents

What are exponents?
Exponents are a shorthand used for repeated multiplication. Remember that when you were first introduced to multiplication it was as a shorthand for repeated addition. For example, you learned that: 4 &times; 5 = 5 + 5 + 5 + 5. The expression "4 &times; " told us how many times we needed to add. Exponents are the same type of shorthand for multiplication. Exponents are written in superscript (that is, a smaller number written above) after a regular-sized number.

For example: 23 = 2 &times; 2 &times; 2 = 8. The number in larger font is called the base. The number in superscript is called the exponent. The exponent tells us how many times the base is multiplied by itself. In this example, 2 is called the base and 3 is called the exponent.

The expression 23 is read aloud as "2 raised to the third power", or simply "2 cubed".

Here are some other examples:

6 &times; 6 = 62 (This would read aloud as "six times six is six raised to the second power" or more simply "six times six is six squared".) 7 &times; 7 &times; 7 &times; 7 = 74 (This would read aloud as "seven times seven times seven times seven equals seven raised to the fourth power". There are no alternate expression for  raised to the fourth power.  It is only the second and third powers that usually get abbreviated because they come up more often.  When it is clear what is being talked about, people often drop the words "raised" and "power" and might simply say "seven to the fourth".)

Basic Rules of Exponents
Multiplying Powers with the Same Base

$$a^{n} \cdot a^{m}=a^{n+m}$$

$$a^{n}$$ means that you have $$a$$ a factor of $$n$$ times. If you add $$m$$ more factors of $$a$$ then you have $$n+m$$ factors of $$a$$.

 Dividing Powers with the Same Base 

$$\frac{a^{n}}{a^{m}}= \frac{\overbrace{a \cdot a \cdot a \cdot ... \cdot a}^\text{n factors of a}}{\underbrace{a \cdot a \cdot a \cdot ... \cdot a}_\text{m factors of a}}=a^{n-m}$$

In the same way that $$a^{n} \cdot a^{m}=a^{n+m}$$ because you are adding on factors of $$a$$, dividing is taking away factors of $$a$$. If you have m factors of a in the denominator, then you can cross out m factors from the numerator. If there were n factors in the numerator, now you have n-m factors in the numerator.

 Raising a Power to a Power 

$$ (a^n)^{m}=a^{n \cdot m}$$

If you think about an exponent as telling you that you have so many factors of the base, then $$(a^n)^m$$ means that you have $$m$$ factors of $$a^{n}$$. So you have m groups of $$a^{n}$$ and each one of those has $$n$$ groups of $$a$$. So you have $$m$$ groups of $$n$$ groups of $$a$$. So you have $$n \cdot m$$ groups of a, or $$a^{n \cdot m}$$

 Products raised to powers 

$$(ab)^n = a^n b^n$$

You can multiply numbers in any order you please. Instead of multiplying together n factors equal to ab, you could multiply all of the a 's together, multiply all the b 's together, then finish by multiplying $$a^n$$ times $$b^n$$

 Quotients raised to powers 

$$\left( \frac{a}{b} \right)^{n}=\frac{a^n}{b^n}$$

You can raise a fraction to a power by raising both the numerator and the denominator to the power.

 Any Nonzero number Raised to the Zero Power Is One 

$$a \ne 0 \implies a^{0} = 1$$

This all means that as long as the base is not zero, when you have an exponent of zero, the expression is always equal to 1.

Proof:

$$\ \frac{a^n}{a^n} = a^{n - n} = a^0 = 1 \ ,\ a \ne 0$$

Note that $$0^0$$ is undefined.

 Negative Exponents 

A negative sign on an exponent means that you need to take the reciprocal of the base, then make the exponent positive.

$$\left( \frac{a}{b} \right)^{-n}=\left( \frac{b}{a} \right)^{n}$$

Examples of Basic Rules of Exponents
Multiplying Powers with the Same Base

Add the exponents:

$$3^{4} \cdot 3^{2}= (3 \cdot 3\cdot 3\cdot3) \cdot (3\cdot3)=3 \cdot 3\cdot 3\cdot3 \cdot 3\cdot3=3^{6}$$

 Dividing Powers with the Same Base 

Subtract the exponents:

$$\frac{5^{7}}{5^{4}}= \frac{5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5}{5 \cdot 5 \cdot 5 \cdot 5}=\frac{5 \cdot 5 \cdot 5}{1}=5^{3}$$

 Raising a Power to a Power 

Multiply the exponents:

$$(4^{3})^{2}=4^{3} \cdot 4^{3} = (4 \cdot 4 \cdot 4) \cdot (4 \cdot 4 \cdot 4)=4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4=4^{6}$$

 Any Nonzero number Raised to the Zero Power Is One 

$$2^{0}=1$$

$$109^{0}=1$$

$$-92^{0}=-1$$

$$(-92)^{0}=1$$

 Exponent Problems 

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 $$1.574 \cdot 10^{15}$$. To convert from regular notation to scientific notation, find the leftmost non-zero digit. Count how many places away it is from the ones digit. This is the exponent for 10. 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. 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 $$\cdot 10^{\text{exponent}}$$. You're done!

Mathematical Insights
1. Motivation for the negative exponent rule:

Since positive exponents are defined by repeated multiplication, see if you can show that if $$a^{-b} = \frac{1}{a^b}$$ then $$ a^b a^c = a^{b + c}$$ for all b and c. Can you think of a different meaning for a negative exponent? Would the operators + and - still have the same meaning?

2. Motivation for the non-definition of $$0^0$$:

Try to define $$0^0$$. A good definition would ensure that $$ a^b a^c = a^{b + c}$$. Show that there is no good definition. (You may find it useful that it is impossible to define 1/0 consistently—why?) Read the article zero. What happened in world history as people moved from the Roman numerals to Hindu-Arabic numerals?

3. The importance of the product rule:

Use examples to show that $$b^0 = 1$$ and $${(a^b)}^c = a^{bc}$$ are consequences of the product rule.

4. Fractional powers:

What would be a sensible definition of $$a^{1/2}$$? $$ a^{m/n} $$?