P.2 Exponents and Scientific Notation

Definition of a Natural Number Exponent

• If b is a real number and n is a natural number,

bn b b b ... b• bn is read “the nth power of b” or “ b to the nth power.” Thus, the nth

power of b is defined as the product of n factors of b. Furthermore, b1 = b

• “Special” powers are 2 (squared)

and 3 (cubed).

Find the exponent button on your calculator.

The Negative Exponent Rule

• If b is any real number other than 0 and n is a natural number, then

b n 1

bn

The negative exponent “flips” the base to the other side of the division bar to become a positive exponent. It DOES NOT CHANGE the SIGN of the base!

Ex: -2

3x-5 ans:

52

3

x

The Zero Exponent Rule

• If b is any real number other than 0,b0 = 1.

Ex: (for x not equal to zero)79 4378 0(56892302 ) 1x

The Product Ruleb m · b n = b m+n

When multiplying exponential expressions with the same base, add the exponents. Use this sum as the exponent of the common base.

Hint: if you get these rules confused, think of a simple example and work it manually. Such as:

2 3 5 (2 3) or x x xx xxx xxxxx x x

The Power Rule (Powers to Powers)(bm)n = bm•n

When an exponential expression is raised to a power, multiply the exponents. Place the product of the exponents on the base and remove the parentheses.

Hint: if you get these rules confused, think of a simple example and work it manually. Such as:

2 3 2 2 2

6 (2 3)

( ) ( ) ( ) ( )

or

x x x x xx xx xx

xxxxxx x x

The Quotient Rule

• When dividing exponential expressions with the same nonzero base, subtract the exponent in the denominator from the exponent in the numerator. Use this difference as the exponent of the common base. (Shortcut: subtract “up” or “down” depending on which is the smaller exponent.)

Hint: if you get these rules confused, think of a simple example and work it manually. Such as:

Q: Can you also think of an example that would demonstrate the zero exponent rule using the quotient rule?

bm

bn bm n

5

2

3 (5 2)

(two x's divide out)

=x or

x xxxxxxxx

x xx

x

Ex: Find the quotient

a)

b) 14x7

10x10

3

2

4

4

Ans: a) b) 5

1 1 or

4 1024 3

7

5x

Products to Powers(ab)n = anbn

When a product is raised to a power, raise each factor to the power.

Hint: if you get these rules confused, think of a simple example and work it manually.

Simplify: (-2y)4.

Long way: (-2y)4 =(-2y)(-2y)(-2y)(-2y)=(-2)(-2)(-2)(-2)(y)(y)(y)(y)

(by commutative law)= (-2)4y4

= 16y4

Short way: (-2y)4 = (-2)4y4 (by “products of powers”)

= 16y4

Example

Solution

Now you try to simplify each of the following, then check below. (Hint: one of these cannot use any of the shortcut rules we have discussed, why?):

-(-3x2y5)4

(x+2) 2

Ans: and

The second one has ADDITION, our rules refer to mult or divisn.

8 2081x y 2 24 4 NOT 4x x x

Quotients to Powers

• When a quotient is raised to a power, raise the numerator to that power and divide by the denominator to that power.

n

nn

b

a

b

a

Example• Simplify by raising the quotient (15x7/6)-4 to

the given power.

4715

6

x

Solution:

Ans: (Hint: reduce inside parenthesis first!) 28

16

625x

Properties of Exponents

1. b n 1

bn 2. b0 1 3. bm bn bmn 4. (bm)n bmn

5.bm

bn bm n 6. (ab)n anbn 7.a

b

n

an

bn

Do problem #62 p 22.

314 8

17 2

30

10

a b

a b

Ans: (Again, inside parenthesis first)

310 30

3 9

3 27b b

a a

The number 5.5 x 1012 is written in a form called scientific notation. A number in scientific notation is expressed as a number greater than or equal to 1 and less than 10 multiplied by some power of 10. It is customary to use the multiplication symbol, x, rather than a dot in scientific notation.

(optional) Scientific Notation

Example• Write each number in decimal notation:

a. 2.6 X 107 b. 1.016 X 10-8

a. 2.6 x 107 can be expressed in decimal notation by moving the

decimal point in 2.6 seven places to the _______.

2.6 x 107 = ________________________.

Solution:

b. 1.016 x 10-8 can be expressed in decimal notation by moving the decimal point in 1.016 eight places to the _______.

1.016 x 10-8 = _______________________.

To convert from decimal notation to scientific notation, we reverse the procedure.• Move the decimal point in the given number to obtain a number greater than

or equal to 1 and less than 10.• The number of places the decimal point moves gives the exponent on 10; the

exponent is positive if the given number is greater than 10 and negative if the given number is between 0 and 1.

Example: Write each number in scientific notation.

a. 4,600,000 ans:

b. 0.00023 ans:

Scientific Notation

64.6 x 10

-42.3 x 10