ExponentsExponentsExponentsExponents

The mathematician’s shorthandThe mathematician’s shorthand

Is there a simpler way to write5 + 5 + 5 + 5?

4 · 5

Just as repeated addition can be simplified by multiplication, repeated multiplication can be simplified by using exponents. For example:

2 · 2 · 2 is the same as 2³, since there are three 2’s being multiplied together.

Likewise, 5 · 5 · 5 · 5 = 54, because there are four 5’s being multiplied together.

Power – a number produced by raising a base to an exponent. (the term 27 is called a power.)

Exponential form – a number written with a base and an exponent. (23)

Exponent – the number that indicates how many times the base is used as a factor. (27)

Base – when a number is being raised to a power, the number being used as a factor. (27)

Evaluating exponents is the second step in the order of operations. The sign rules

for multiplication still apply.

Writing exponents

3 · 3 · 3 · 3 · 3 · 3 = 36 How many times is 3 used as a factor?

(-2)(-2)(-2)(-2) = (-2)4 How many times is -2 used as a factor?

x · x · x · x · x = x5 How many times is x used as a factor?

12 = 121 How many times is 12 used as a factor?

36 is read as “3 to the 6th power.”

Evaluating Powers

26 = 2 · 2 · 2 · 2 · 2 · 2 = 6483 = 8 · 8 · 8 = 51254 = 5 · 5 · 5 · 5 = 625Always use parentheses to raise a negative

number to a power.(-8)2 = (-8)(-8) = 64(-5)3 = (-5)(-5)(-5) = -125(-3)5 = (-3)(-3)(-3)(-3)(-3) = -243

When we multiply negative numbers together, we must use parentheses to

switch to exponent notation.

(-3)(-3)(-3)(-3)(-3)(-3) = (-3)6 = 729

You must be careful with negative signs!

(-3)6 and -36 mean something entirely different.

Note:When dealing with negative numbers, *if the exponent is an even number the

answer will be positive.

(-3)(-3)(-3)(-3) = (-3)4 = 81

*if the exponent is an odd number the answer will be negative.

(-3)(-3)(-3)(-3)(-3) = (-3)5 = -243

In general, the format for using exponents is:

(base)exponent

where the exponent tells you how many times the base is being multiplied together.

Just a note about zero exponents: powers such as 20, 80 are all equal to 1. You will learn more about

zero powers in properties of exponents and algebra.

Simplifying Expressions Containing Powers

• Simplify 50 – 2(3 · 23)50 – 2(3 · 23)= 50 – 2(3 · 8) Evaluate the exponent.

= 50 – 2(24) Multiply inside parentheses.

= 50 – 48 Multiply from left to right.

= 2 Subtract from left to right.

Problem Solving

Many problems can be solved by using formulas that contain exponents. Solve the problem below:

The distance in feet traveled by a falling object is given by the formula d = 16t2, where t is the time in seconds. Find the distance an object falls in 4 seconds.

Problem Solving

The sum of the first n positive integers is ½(n2 + n). Check the formula for the first 4 positive integers. Then use the formula to find the sum of the first 12 positive integers. 1 + 2 + 3 + 4 = 10

½(n2 + n)½(122 + 12)½(144 + 12)

½(156)78

Simplify and Solve

1) (3 - 62) =

2) 42 + (3 · 42)

3) 27 + (2 · 52)

4) (-3)5

5) 2(53 + 102)

• A population of bacteria doubles in size every minute. The number of bacteria after 5 minutes is 15(25). How many bacteria are there after 5 minutes?

Properties of Properties of ExponentsExponents

Properties of Properties of ExponentsExponents

Multiplying, dividing powers and Multiplying, dividing powers and zero power.zero power.

The factors of a power, such as 74, can be grouped in different ways. Notice the relationship of the exponents in each

product.

7 · 7 · 7 · 7 = 74

(7 · 7 · 7) · 7 = 73 · 71 = 74

(7 · 7) · (7 · 7) = 72 · 72 = 74

Multiplying Powers with the Same Base

• To multiply powers with the same base, keep the base and add the exponents.

• 35 · 38 = 35+8 = 313

• am · an = a m+n

Multiply

• 35 · 32 = 35+2 = 37

• a10 · a10 = a10+10 = a20

• 16 · 167 = 161+7 = 168

• 64 · 44 = Cannot combine; the bases are not the same.

Dividing Powers with the Same Base

• To divide powers with the same base, keep the base and subtract the exponents.

• 69 = 69-4 = 65

64

• bm = bm-n

bn

Divide

• 1009 = 1009-3 = 1006

1003

• x8 = Cannot combine; the bases are not the same.

y5

When the numerator and denominator of a fraction have the same base and exponent, subtracting the exponents results in a 0 exponent.

1 = 42 = 42-2 = 40 = 1 42

•The zero power of any number except 0 equals 1.

1000 = 1

(-7)0 = 1

a0 = 1 if a ≠ 0

How much is a googol?

10100

Life comes at you fast, doesn’t it?

Negative ExponentsNegative ExponentsNegative ExponentsNegative Exponents

Extremely small numbersExtremely small numbers

Negative exponents have a special meaning. The rule is as follows:

Basenegative exponent =

Base1/positive exponent

4-1 = 1 41

Look for a pattern in the table below to extend what you know about exponents. Start with what you know about positive

and zero exponents.

103 = 10 · 10 · 10 = 1000102 = 10 · 10 = 100101 = 10 = 10100 = 1 = 110-1 = 1/1010-2 = 1/10 · 10 = 1/10010-3 = 1/10 · 10 · 10 = 1/1000

Example:

10-5 = 1/105 = 1/10·10·10·10·10 = 1/100,000 = 0.00001

So how long is 10-5 meters?10-5 = 1/100,000 = “one hundred-thousandth of a meter.

Negative exponent – a power with a negative exponent equals 1 ÷ that power with a positive exponent.

5-3 = 1/53 = 1/5·5·5 = 1/125

Evaluating negative exponents

1) (-2)-3 = 1/(-2)3 = 1/(-2)(-2)(-2) = -1/8

2) 5-3 = 1/53 = 1/(5)(5)(5) = 1/ 125

3) (-10)-3 = 1/(-10)3 = 1/(-10)(-10)(-10) = -1/1000 = 0.0001

4) 3-4 · 35 = 3-4+5 = 31 = 3 Remember Properties of Exponents: multiply same base you keep the base and add the exponents.

Evaluate exponents:Get your pencil and calculator ready to

solve these expressions.

1) 10-5 =

2) 105 =

3) (-6)-2 =

4) 124/126 =

5) 12-3 · 126

6) x9/x2 =

7) (-2)-1 =

8) 23/25 =

Problem Solving using exponents

The weight of 107 dust particles is 1 gram. How many dust particles are in 1 gram?

As of 2001, only 106 rural homes in the US had broadband internet access. How many homes had broadband internet access?

Atomic clocks measure time in microseconds. A microsecond is 0.000001 second. Write this number using a power of 10.

Exponents can be very useful for

evaluating expressions,

especially if you learn how to use your

calculator to work with them.