Section 6: Exponents and Roots

6.1 Exponents

Definition 12 For the term an where a is a real number and n is a whole number, the a is called the base and the n is called the exponent. It means to multiply a by itself n times.

The definition of exponents can be extended beyond whole numbers to any rational number. However, for basic probability and statistics, whole numbers are sufficient.

Example 16 x4 = xxxx
Example 17 43 = 4(4)(4) = 48
Example 18 (0.5)2 = (0.5)(0.5) = 0.25

The basic rules for exponents are used for multiplying and dividing with a common base. Let's look at a couple of examples before we state the rules.

Example 19 x2(x7) =xx(xxxxxxx) =x9
Example 20 d3(d4) = ddd(dddd) = d7
Example 21 x5/x2 = (xxxxx)/(xx) = xxx = x3
Example 22 (y2)3 = (y2)(y2)(y2) = (yy)(yy)(yy) = y6

Perhaps you can guess the rules from the examples. Notice that the base must be the same for all three rules to apply. They are:

• anam = an+m
• (an)m = anm
• an/am = an-m

Also, there is the special case of a zero exponent. To determine what a zero exponent means, we look back at the division rule:

• an/am = an-m

Consider 23/22. The rule says this is 23-2 = 21 = 2. What if it had been 23/23 instead? We know that this is 8/8 = 1. But the rule says it would be 23-3 = 20. Therefore, to be consistent, 20 must be 1. (And it is!) In fact, the zero exponent rule is:

• a0 = 1 for any non-zero value of a.

Any exponent problems that you would encounter in basic probability and statistics can be computed by the yx key on a calculator. As mentioned above, the exponent can be any rational number (positive or negative). These also be computed by the yx calculator key.

6.2 Roots

A concept similar to exponents is roots. The most common root is the square root. We already mentioned square roots when we introduced irrational numbers in section 1.1. A square root of any number, n, is the number that you multiply by itself to get n. For example the square root of 25 is 5, the square root of 16 is 4, and the square root of 1,849 is 43. (I bet you didn't know that last oue did you?). While it's true that -5 multiplied by itself will also give you 25, we generally use the positive or principle root. The symbol for square roots is √n

Of course there are other roots. The cubed root of any number, n, is the number you multiply by itself three times to get n. For example, the cubed root of 8 is 2 because 2 × 2 × 2 = 8. Similarly3 × 3 × 3 = 27 so the cubed root of 27 is 3. The notation for cubed roots is 3√n.

In reality, there are an infinite number of roots. For example: As with exponents, most calculators have special root keys to compute these for you. However in basic probability and statistics, you will usually encounter square roots and maybe an occasional cubed root. There are only a couple of rules you should keep iu mind:

• Whenever you are seeking a even root (square, 4th, 6th, etc.) of a number, the number must be positive. There is no real number that can be multiplied by itself an even number of times to get a negative number.
• Whenever you are seeking au odd root (cubed, 5th, 7th, etc.) of a number, the root will have the same sign (positive or negative) as the number.

6.3 Order of Operations Revisited

The basic order of operations werepresented in section 4. Now we need to insert exponents in the appropriate place:

1. Perform all operations inside symbols ofinclusion (parentheses, etc.) starting with the innermost set of symbols and working outward. Within inclusion symbols, follow the other rules. Remember that fraction bars imply inclusion symbols.
2. Evaluate exponents and roots.
3. Perform all multiplications and divisions in the order they occur reading from left to right.
4. Perform all additions and subtractions in the order they occur reading from left to right.