Lesson #8 Exponents: Rules and the Order of Operations

Exponents are a notational system for marking the multiplication of several like factors. It gets hard to write a long multiplication problem like 2(2)(2)(2)(2)(2)(2)(2)(2)(2). If we miscount the number of times to multiply by 2, it can be easy to come up with the wrong answer (multiplying 10 copies of 2 should be 1024). Even worse, there are times we deal with even larger products, where the exact answer is longer than even a calculator can display. (What is the product of 100 copies of 2 all multiplied together? 2¹⁰⁰ is too big to write out its digits in any understandable way.)

We will deal with this through exponent notation. The multiplication of 10 copies of 2 is written as 2¹⁰. The base number (regular size, in normal type position) is the factor to use in the multiplication. The exponent (smaller size, raised above and to the right of the base) indicates how many copies to use. The exponent always refers to the base number below to its left. Read 2¹⁰ as "two to the tenth power" or "two to the power ten". The exponent is an operation, giving you an instruction for how to carry out the work on two number symbols. Remember, the exponent counts the number of times to multiply the base by itself.

Example 1. Simplify each of the following.

(a) 3². This means "multiply two copies of three": 3× 3 = 9. So 3² = 9.

(b) 5³ means 5× 5× 5 = 25× 5 = 125. (Multiply 3 copies of 5).

(c) (-4)² means (-4)(-4) = +16. (Multiply 2 copies of (-4); note that the exponent is over the ")", and therefore applies to the entire contents of the parentheses). The answer is positive since negative times negative is positive.

(d) (-2)³ means (-2)(-2)(-2) = (+4)(-2) = -8. Multiplying three copies of (-2) yields a negative answer.

Since exponents are actually referring to the operation of multiplication, they must be carried out before any other multiplications (it's like deciphering a code before you can read the message). So the full Order of Operations for doing arithmetic is the following:

Order of Operations

1) Do anything inside parentheses first.

2) Do any work using exponents.

3) Do the multiplication and division.

4) Do the addition and subtraction

Always follow this order. If more than one operation of the same type is given, do these in order from left to right. In a fraction, you must also complete any calculations in the numerator or the denominator separately before trying to reduce the fraction.

 Example 2. Simplify each of the following.

(a) .		First do the calculation inside the parentheses.
	4(9) - 8	Then do the exponents. (3)^2 = 9, and 2^3 = 8. Note that we keep the parentheses around the 9 since the multiplication is not yet finished.
	36 - 8	The multiplication is next.
	28	Finish by doing the subtraction.
(b) .		With more than one type of parentheses, start with the innermost group first, and work your way out.
		Inside the [ ], the exponent is done before the multiplication and the subtraction.
	-2	Everything inside the [ ] reduced to just -5. Finish the problem by then doing any operations left outside the parentheses.
(c) .		Start by doing the top and bottom separately. The top multiplication and the bottom exponent are done before the subtractions.
		Then do the subtractions, and reduce the fraction in the end.

With variables, exponents are especially important in our notation. The meaning, of course, is the same. x⁴ means x× x× x× x (multiply four copies of whatever x is). We will need a couple rules for simplifying exponents on variables. Each rule helps to simplify a term with more than one use of a variable to one letter with a single exponent.

Multiplying the same base: x³× x⁴ = x³⁺⁴ = x⁷ Add exponents to multiply the same base.

In detail: x^3× x⁴ = xxx× xxxx = multiply 7 copies of x = x⁷. The exponent counts the total number of factors of x used in the multiplication.

Dividing the same base: y⁵/y² = y³ Subtract exponents to divide the same base.

In detail: y⁵/y² = = yyy (cancel two factors of y). Since division is the opposite of multiplication, the total number of factors of y is reduced (subtract) by the number of factors in the denominator.

Powers to an exponent: (A³)⁴ = A¹² Multiply exponents over exponents.

In detail: (A³)⁴ means A³× A³× A³× A³ = A¹² (multiply 4 copies of A³). Again, we are counting the total number of factors of A. Since we have A³ four times, we multiply the exponents.

Two notes on notation: (1) a variable without an exponent can be viewed as having an exponent of 1. So x means x¹. (2) The zero power on any number or variable always gives the value 1. That is, x⁰ means "multiply no copies of x"---only the unseen coefficient 1 is left.

Exercises

Simplify each of the exponent operations.

1) 4^2	2) 2^4	3) (-5)^2	4) 3^3	5) (-2)^3
6) (-8)^2	7) -4^3	8) (-4)^3	9) 1^7	10) (-1)^6
11) 5^0	12) (-3)^0	13) 0^2	14) -3^2	15) -2^5

Simplify by using the Order of Operations.

16) 3× 2 + 2	17) 2 + 5 - 3× 2	18) 5× 4 - 2× 8
19) 2^3 - 8 ¸ 2	20) 9 + 16 ¸ 2^3	21) -2× 4^2
22) (-3)^2 - 2	23) -3^2 - 2	24) 10 - 4^2
25) (3 + 4× 3^2) ¸ (2× 8 - 5 - 2^3)	26)	27)
28)	29)	30) 3[4(6 - 2) + 1] - 7
31) 2[2(6 + 2) ¸ 4] + 1	32) (4 - 8)(10 - 7)	33) 4× 5 + 8 ¸ 2^2 - 2× 5
34)	35)	36)
37)	38)	39)
40)	41)	42)
43) 18 ¸ 3^2 + 4× 2 - 1	44) 3^2 - 5^2	45) 28 - 2^3(3^2 - 3)

Evaluate each expression when a = 2 and b = -3.

46) a^2 - b	47) 4b^2	48) 3a^2 + 7a - 12	49) 2 + 3(a - b)^2
50) b^2 + 4b - 1	51) -b^2	52) a^2b + 3b	53) 5 - 2b + 5a^2

Simplify the following using exponent rules.

54) x^3× x^2	55) y× y^5	56) x^2× x^4	57) a^6× a^2	58) t^3× t
59) y^9/y^5	60) x^5/x^3	61) a^8 ¸ a^2	62) b^3 ¸ b	63) w^4/w^3
64) x^3/x^3	65) t^0	66) y^2 ¸y^2	67) x× x	68) b× b^2
69) (x^2)^3	70) (a^5)^2	71) (y^3)^3	72) x^3× x^7	73) x^4/x

Scientific Notation

Exponents can also be used to write very large and very small numbers in a more concise way. Scientific Notation is one common way used in many science applications. To write a number in Scientific Notation, move the decimal place far enough that there is exactly one (nonzero) digit left of the decimal point. Then multiply by a power of 10, where the exponent is the number of places you moved the decimal point (use positive exponent if the decimal moved left, and negative exponent if the decimal moved right). This works to represent the same number, because multiplying by a power of 10 simply moves the decimal point back to where it belongs.

Example 3.

45,000,000 is the same as 4.5 10⁷ Note: it is standard to use "" for "times" here.

7,810,000,000 = 7.8110⁹. Notation is simplified because there is no need to write a long string of 0 digits at the end of a number.

540 = 5.4 10²

0.0000003 = 3 10^-7 Negative exponents give numbers less than 1. Essentially, 3 10^-7 means 3 ¸ 10⁷ (remember, division is the opposite of multiplication).

0.000764 = 7.64 10^-4

Example 4. Using Scientific Notation can simplify calculations involving large numbers. First, convert numbers to Scientific Notation, and then use exponent rules to reduce the powers of 10 to a single exponent.

Simplify (320,000)(50,000,000).

Solution:

(320,000)(50,000,000)		(convert notation)
		(multiply coefficients & add exponents on 10)
	= 16,000,000,000,000	(move decimal point back 12 places)

More Exercises

Write each number in standard notation.

74) 610^5	75) 1.210^8	76) 3.5110^-4
77) 8.10510^13	78) 1.0710^-8	79) 9 10^3

Write each number in scientific notation.

80) 55,000	81) 480,000,000	82) 0.0000127
83) 0.0004	84) 200,000	85) 1,035,000,000,000

Convert each number to scientific notation, and then simplify using the exponent rules.

86) (500)(30,000)

87) (120,000)(3,000,000)

88) (350)(20,000,000)

Check the answers to Lesson 8 exercises

Go on to Lesson 9

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