Scientific notation allows you to express very large and very small numbers without using large numbers of digits and decimal places.  It's all done with powers of ten. Here are some examples that clearly show this fact:

• A very large number such as 2,000,000,000 can be written with scientific notation as 2 x 10⁹
• A very small number such as 0.000000674 can be written with scientific notation as 6.74 x 10^-7

What does an expression such as  2 x 10⁹ mean? Well, if you understand your exponents, you can figure that 10⁹ (ten to the ninth power) is the same as 1,000,000,000. So 2 x 10⁹ is the same as 2 x 1,000,000,000. And that is the same as 2,000,000,000. So 2 x 10⁹ is the same as 2,000,000,000, and vice versa. Or to be very "mathematical" about it:

2 x 10⁹ = 2,000,000,000

In the second example, the 10 has an exponent of – 7. This would a rather small number: Recall that from your earlier studies of numbers having negative exponents:. This means that  6.74 x 10^-7 is the same as . And if you do the division, you can see that . To make a long story short, you should be able to see that 6.74 x 10^-7 is the same as 0.000000674.

6.74 x 10^-7 = 0.000000674

Try looking at it this way:

• 2 x 10³ is the same as 2 x 1000, or 2000
• 4.5 x 10⁶ is the same as 4.5 x 1,000,000, or 4,500,000
• 8 x 10^-3 is the same as 8 x .001, or 0.008

Examples

Are the the following  values expressed in proper scientific notation? Explain your response.

2.4 x 102	Yes, because there is exactly one non-zero digit to the left of the decimal point.
67.48 x 104	No, because there is more than one digit to the left of the decimal point.
– 4.5 x 106	Yes, because there is exactly one non-zero digit to the left of the decimal point.
0.25 x 103	No, because the digit to the left of the decimal point is a zero.

Rewriting Decimal Values in Scientific Notation

You should be able to write any decimal value in scientific notation.

Step 1. Determine how many times the decimal value has to be multiplied or divided by 10 in order to end up with exactly one digit to the left of the decimal point.

How many times does the number 54782 have to be divided by 10 in order to take this normalized form, 5.47825?. Here is the how that is done:

54782 ÷ 10 = 5478.2

5478.2 ÷ 10 = 547.82

547.82 ÷ 10 = 54.782

54.782 ÷ 10 = 5.4782

So you have to divide 54782 by 10 — four times — in order to get the proper form for scientific notation.

But you can't simply divide a term by some multiple of 10,  and then quit. You have radically changed the value or the original decimal. You can use a power-of-10 to restore the value.

Step 2. Use a power-of-10 expression to restore the value of the original decimal.

Recall in this example that you divided 54782 by ten 4 times to get 5.4782. Now you have to multiply  by 10⁴ to restore the original value.

So 54782 can be rewritten as 5.4782 x 10⁴

Examples:  Normalizing the Coefficient

1. How many times does 1234 have to be divided by 10 in order to become 1.234?  Ans: 3 times
2. How many times does 6880000 have to be divided by 10 in order to become 6.880000? Ans: 6 times
3. How many times does 0.025 have to be multiplied by 10 in order to become 2.5?  Ans: 2 times
4. How many times does 0.0000005 have to be multiplied by 10 in order to become 5.0? 7 times
5. How many times does 256.2 have to be divided my 10 in order to become 2.562?  2 times

Examples: Determining the Power of Ten

1. if you divided a decimal by ten 4 times, the corresponding power of ten is  ____. Ans: 10⁴
2. If you multiplied multiplied a decimal by ten 12 times, the corresponding power of ten is ____. Ans: 10^–12

Examples and Exercises

Rewriting Decimal Values in  Powers of 10

Looking at powers of ten the other way around, you should be able to see that:

600,000 can be rewritten as 6 x 10⁵

7,230 can be rewritten as 7.23 x 10³

0.00034 can be rewritten as 3.4 x 10^-4

There are two important things to bear in mind:

• First, the base of the exponent is always 10.
• Second, the whole number part of the coefficient must be a single digit between –9 and +9.

Examples

Rewrite the following decimal values in the proper scientific form (one, and only one digit, on the left side of the decimal point.

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Examples and Exercises

Changing the Location of a Decimal Point and the Value of the Exponent

When working with powers of ten and scientific notation it is often necessary to adjust the position of the decimal point in the coefficient or to change the value of the exponent. When changing one of these terms, it is important that you change the other in order to keep the overall value the same.

Adjusting the Decimal Point

When working with scientific notation, you are often required to change the location of the decimal point in the coefficient, but when you move the decimal point, you must adjust the value of the coefficient. Consider all these different ways to write 1235:

Examples

Rewrite 78.5 x 10² after moving the decimal point one place to the left.  Ans: 7.85 x 10³

Rewrite 120000 x 10¹² after moving the decimal point four places to the left. Ans: 12 x 10¹⁶

Examples & Exercises

There are also many instances where it is necessary change the value of the exponent in a power-of-10 term.  When increasing or decreasing the value of the exponent, you must change the location of the decimal point in the coefficient.

Examples

Rewrite 78.5 x 10² after changing the exponent to 3. Ans: 7.85 x 10³

Rewrite 120000 x 10¹² after changing the exponent to 16. Ans: 12 x 10¹⁶

Examples & Exercises