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Title:

Hotmath Algebra 1

Author:

Hotmath Team

Free, Lessons

Chapter:Exponents, Polynomials and FactoringSection:Scientific Notation

Section 6-2

Scientific Notation

Which is bigger: 391000000000000000000 or 86400000000000000000?

To tell, you have to count all those zeros. Unless you have really good eyes, it will probably give you a headache.

Scientific notation was developed by scientists who use really really big numbers and really really small numbers all the time, and were sick of getting headaches from counting lots of zeros.

The first thing to remember is your powers of 10. (If you're confused by this table, see the pages on exponents and the properties of exponents.)

Powers of 10
10^–5 = 0.00001	10¹ = 10
10^–4 = 0.0001	10²= 100
10^–3 = 0.001	10³ = 1,000
10^–2 = 0.01	10⁴ = 10,000
10^–1 = 0.1	10⁵ = 100,000
10⁰ = 1	10⁶ = 1,000,000

For positive powers of 10 , the exponent is the same as the number of zeros after the 1. The negative powers of 10 show how many places there are to the right of the decimal point.

The idea behind scientific notation is to write numbers as a product of a normal-sized number, and a power of 10.

If you counted the big numbers at the beginning of the section, you found that 391000000000000000000 has 18 zeros. So you can write it as

391 × 10¹⁸

Much easier to read! But, to make scientific notation standard, there is a convention that the first number in the product should be greater than or equal to 1, and less than 10. So, we divide 391 by 100 (or 10²) to get 3.91. Then we make up for it by multiplying the second number by 10². So, we end up with the number in scientific notation:

3.91 × 10²⁰

The other big number at the top of the page was 8.64 × 10¹⁹. When the numbers are written in scientific notation it's much easier to compare them and do calculations.

The same thing works for small numbers, like 0.000076. First move the decimal point five points to the right to get 7.6 (which is between 1 and 10). To compensate, multiply by 10^–5:
7.6 × 10^–5

Problem: 1

Express 5 10⁵ in standard notation.

Problem: 3

Express 3.7 10³ in standard notation.

Problem: 5

Express 5.18 10⁷ in standard notation.

Problem: 7

Express 7 10^–6 in standard notation.

Problem: 9

Express 6.7 10^–3 in standard notation.

Problem: 11

Use standard notation to write 6.425 10³.

Problem: 13

Use standard notation to write 1.45 10^–3.

Problem: 15

Use standard notation to write 6.8 10⁴.

Problem: 17

Use standard notation to write 1.7 10^–5.

Problem: 19

Use standard notation to write 8 10^–4.

Problem: 21

Use standard notation to write 1.008 10^–3.

Problem: 23

Use scientific notation to write 15,800.

Problem: 25

Use scientific notation to write 0.067.

Problem: 27

Use scientific notation to write 625,000.

Problem: 29

Use scientific notation to write 6,400,000.

Problem: 31

Use scientific notation to write 0.0000078.

Problem: 33

Express 33,000,000 in scientific notation.

Problem: 35

Simplify.

6.2 10^–5· 4.1 10⁸

Problem: 37

Multiply and express the result in scientific notation.

(5 10⁵)(3 10³)

Problem: 39

Multiply and express the result in scientific notation.

(5.0 10⁸)(7.0 10⁴)

Problem: 41

Multiply and express the result in scientific notation.

(2.4 10^–4)(5.0 10^–9)

Problem: 43

Divide and express the result in scientific notation.

Problem: 45

Divide and express the result in scientific notation.

Problem: 47

Divide and express the result in scientific notation.

Problem: 49

Divide and express the result in scientific and standard notation.

Problem: 51

The mass of a helium atom is 0.000000000000000000000007 gram. Write this number in scientific notation.

Problem: 53

Light travels through space at about 3.0 10⁵ km per second. Suppose it takes 25 10² seconds for light from the sun to reach an asteroid out somewhere between the orbits of Mars and Jupiter. How far away is the asteroid?