Lesson 21 Section 2
Exponent 0
Any number (except 0) with exponent 0 is defined as 1.
a0 = 1
Problem 8. Evaluate the following..
To see the answer, pass your mouse over the colored area.
To cover the answer again, click "Refresh" ("Reload").
Do the problem yourself first!
| a) | 50 = 1 | b) | (−5)0 = 1 | c) | −50 = −1 | d) | (½)0 = 1 |
| e) | 3· 100 = 3· 1 = 3 | f) | (3· 10)0 = 1 | g) | −3· 100 = −3· 1 = −3 |
h) 5 + 50(5 + 50) = 5 + 1(5 + 1) = 5 + 6 = 11
Why do we define a0 as 1?
For one thing, we have this rule:
 |
= | am − m | = | a0. | |
| But | |
= | 1. |
Therefore, we must define a0 as 1.
As for 00, that could be any number! For,
 |
= | Any number. | Lesson 6 |
But
 |
= | 00. |
Therefore we must conclude that 00 could be any number.
Problem 9. Use the rules of exponents to evaluate the following.
| a) |  |
= | 107 − 2 + 3 − 8 = 100 = 1 |
| b) |  |
= | 22 − 8 + 4 + 3 − 6 + 5 = 20 = 1 |
Scientific notation
"Scientific notation" has one non-zero digit to the left of the decimal point.
2.345
That number is written in scientific notation. There is one digit to the left of the decimal point -- 2 -- and it is not 0.
In general, a number written in scientific notation will be multiplied by 10 raised to an exponent.
Example 1. 6.45 × 103. This is the scientific notation for what number?
Answer. Since the exponent is positive, we will be multiplying 6.45. To multiply a decimal by a power of 10, move the decimal point right the number of places indicated by the exponent:
6.45 × 103 = 6,450
Move the point three places right. To do that, we must add on a 0.(Skill in Arithmetic, Lesson 3.)
Example 2. 6.45 × 10−3. This is the scientific notation for what number?
Answer. Since the exponent is negative, we will be dividing 6.45. (For,
| 10−3 = | 1 1000 |
.) To divide a decimal by a power of 10, move the decimal |
point left the number of places indicated by the exponent:
6.45 × 10−3 = .00645
Move the point three places left. To do that, we must add on 0's.
Problem 10. Each of these is written in scientific notation. Multiply out.
| a) | 2.401 × 103 = 2,401 | b) | 1.006 × 10² = 100.6 | |
| c) | 7.01 × 105 = 701,000 | d) | 4.986 × 10 = 49.86 | |
| e) | 3.85 × 10−4 = .000385 | f) | 1.8 × 10−1 = .18 | |
| g) | 6.0 × 10−3 = .006 | h) | 4.3 × 10−8 = .000000043 | |
Example 3. Write in scientific notation.
a) 562.4 = 5.624 × 102
To change 562.4 into 5.625, we moved the decimal point two places left. To compensate, we must multiply by 10 with exponent +2.
b) .00395 = 3.95 × 10−3
To change .00395 into 3.95, we moved the decimal point three places right. To compensate, we must multiply by 10 with exponent −3.
Problem 11. Write in scientific notation.
| a) | 1234 = 1.234 × 103 | b) | 12.34 = 1.234 × 10 | |
| c) | 219,000,000 = 2.19 × 108 | d) | 0.000081 = 8.1 × 10−5 | |
| e) | 0.00395 = 3.95 × 10−3 | f) | 0.000000002 = 2.0 × 10−9 | |
Next Lesson: Multiplying and dividing algebraic fractions
Please make a donation to keep TheMathPage online.
Even $1 will help.
Copyright © 2001-2007 Lawrence Spector
Questions or comments?
E-mail: themathpage@nyc.rr.com