Download Section 5.7

Negative Exponents and Scientific Notation - Section 5.7
Consider the following quotient of two exponential expressions (assume b ≠ 0:
b2
b5
Negative Exponents Rule:
If b ≠ 0 and n any natural number, then
b −n = 1n
b
b≠0
Use the negative exponents rule:
x −7
6 −2
5 −3
7 −1
−3 −4
−3 −4
1
Negative exponents in the numerator and denominator (assume b ≠ 0):
−n
b −n = b = 1n
1
b
and
1 = 1 = 1 ⋅ bn = bn = bn
1
1
1
b −n
n
b
Write each expression with positive exponents only. Simplify, if possible.
7 −2
2 −3
4
5
−2
1
2y −3
1
2y −3
2
Simplifying Exponential Expressions:
1. Be sure each base appears only once in your simplified expression using
b m ⋅ b n = b m+n
b m = b m−n
bn
or
2. Remove any parentheses using
ab n = a n b n
a
b
or
n
n
= an
b
3. Simplify any powers using
b m  n = b mn
4. Rewrite exponential expressions with zero powers as 1 (b 0 = 1).
5. Finally, write final result with positive exponents using
−n
b −n = b = 1n
1
b
or
1 = bn = bn
1
b −n
Examples: Simplify
x2
x 10
75x 3
5x 9
50y 8
−25y 14
3
6a 4 
2a 11
x8
x5
2
−4
3y 5  −2 y 4
4
2a −3 b 5 
−4
4x 3 y −2 z
−2x −1 y 5 z 4
5
4x 3 y −2 
4x 2 y
−2
−1
a 3 b 2 c −5
a −3 b 5 c
−2
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Scientific Notation:
The current U.S. debt is about $19.96 trillion (http://www.usdebtclock.org). That amount written in decimal
notation would be $19,960,000,000,000, but we can write it more compactly using what is called scientific notation.
Names of some large numbers:
Number: 10 2 = 100 10 3 = 1, 000 10 6
Name:
hundred
thousand
10 9
10 12
10 15
10 18
10 100
10 googol
million billion trillion quadrillion quintillion googol googolplex
A positive number is written in scientific notation when it is expressed in the form
a × 10 n
where a is a number greater than or equal to 1 and less than 10 (1 ≤ a < 10) and n is an integer.
The current U.S. debt can be written in scientific notation as 19. 96 trillion = 1. 996 × 10 13 .
Consider the following expressions:
2. 1 × 10 3
1. 3 × 10 −2
To convert from scientific notation to decimal notation we use n, the exponent on 10, to move the decimal point.
If n is positive, move the decimal place to the right n places, adding zeros as needed. If n is negative, move the
decimal to the left |n| places, adding zeros as needed.
Write each number in decimal notation:
7. 403 × 10 9
3. 17 × 10 − 6
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Converting from Decimal Notation to Scientific Notation
Write number in the form a × 10 n
1. Determine a, the numerical factor, by moving the decimal place in the number to make a value greater than or
equal to 1 but less than 10.
2. The absolute value of n is the number of places you move the decimal place. The exponent n is positive if the
decimal place was moved to the left and negative if the decimal place was moved to the right.
Write each number in scientific notation:
7,410,000,000
0.000092
Computations with Numbers in Scientific Notation
Multiplication:
a × 10 n  × b × 10 m  = a × b × 10 n + m
Division:
a × 10 n =
b × 10 m
a
b
× 10 n − m
Exponentiation:
a × 10 n  m = a m  × 10 n m
May need to adjust final answer to get it in scientific notation.
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Perform the following operations:
3 × 10 8 4 × 10 5 
20 × 10 9
5 × 10 − 4
2 × 10 3 
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The U.S. national debt is currently $18.4 trillion. The current U.S. population is approximately 318.9 million
people. If we had to pay off our national debt today, how much would we each owe if all the people in the U.S. paid
an equal share? Use scientific notation to work out your solution.
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