Scientific Notation Lesson

Over the course of the last two lessons, we have learned how to work with exponents. One immediate application of integer exponents is the ability to write a number in scientific notation. Scientific Notation is a way to conveniently write very large or very small numbers. Although the process is quite simple, we have to remember a few facts from our pre-algebra studies.
Recall that when multiplying by 10 or a power of 10 we move our decimal point one place to the right for each zero in the power of 10. Let's look at the following example:
9 x 10 » 9.0 x 10 = 90
We move the decimal point one place to the right since there is one zero in 10.
9.23 x 100 » 9.23 x 100 = 923
We move the decimal point two places to the right since there are two zeros in 100.
9.4785 x 1000 » 9.4785 x 1000 = 9478.5
We move the decimal point three places to the right since there are three zeros in 1000.
A similar technique is used when dividing by 10 or a power of 10. We move our decimal point one place to the left for each zero in the power of 10.
89.75 » 10 = 89.75 ÷ 10 = 8.975
We move our decimal point one place to the left since there is one zero in 10.
343,222.55 ÷ 100,000 » 343,222.55 ÷ 100,000 = 3.4322255
We move our decimal point five places to the left since there are five zeros in 100,000.
When we work with 10 raised to a positive integer power, we can write a 1 followed by the exponent number of zeros.
10³ = 1000
1000 is a 1 followed by 3 zeros.
10⁵ = 100,000
100,000 is a 1 followed by 5 zeros.
Let's look at a few examples, and then we will move into writing a number in scientific notation.
Example 1: Find each product.
4.25 x 10³
We know that 10³ is 1000. We also know if we multiply by 1000, we move our decimal point 3 places to the right.
4.25 x 10³ = 4250
Example 2: Find each quotient.
98.35 ÷ 10⁴
We know that 10⁴ is 10,000. We also know if we divide by 10,000, we move our decimal point 4 places to the left.
98.35 ÷ 10⁴ = 0.009835

Scientific Notation

Now that we understand the basic principles that make scientific notation work, let's look at the step-by-step procedure to place a number in scientific notation.

• Place the decimal point to the right of the first (leftmost) non-zero digit in the number
• Count the number of places the decimal point was moved. This will be the absolute value of the exponent on 10
  • The exponent is positive if the original number is larger than the new number formed
  • The exponent is negative if the original number is smaller than the new number formed
• Set up our number in scientific notation as: a x 10ⁿ
  • a represents the number formed in the first step
  • n is our exponent found in step 2

Let's try a few examples.
Example 3: Write each number in scientific notation.
92,000
Step 1) Place the decimal point to the right of the first (leftmost) non-zero digit in the number. In our case, we would place the decimal point after the 9:
9.2000
Step 2) Count the number of places the decimal point was moved. We moved the decimal point 4 places to the left. 4 will be the absolute value of our exponent on 10. Since our original number is larger, we can keep 4 as a positive number.
Step 3) Set up our number in scientific notation as: a x 10ⁿ
a = 9.2
n = 4
92,000 = 9.2 x 10⁴
We could easily check this with multiplication. Multiplying by 10⁴ means we move our decimal point four places to the right. This would give us 92,000 back, therefore, we know our answer is correct.
Example 4: Write each number in scientific notation.
0.0000318
Step 1) Place the decimal point to the right of the first (leftmost) non-zero digit in the number. In our case, we would place the decimal point after the 3:
3.18
Step 2) Count the number of places the decimal point was moved. We moved the decimal point 5 places to the right. 5 will be the absolute value of our exponent on 10. Since our original number is smaller, we need (-5) as our exponent on 10.
Step 3) Set up our number in scientific notation as: a x 10ⁿ
a = 3.18
n = -5
0.0000318 = 3.18 x 10^-5
We can check this in a similar way. We know that 10^-5 is 1/100,000. If we multiply by this number, we are essentially dividing by 100,000. This means we would move our decimal point five places to the left. This would give us 0.0000318 back, therefore, we know our answer is correct.

Multiplying & Dividing in Scientific Notation

We may also be asked to multiply or divide with numbers in scientific notation. Let's look at an example.
Example 5: Find each product.
(3.15 x 10³)(2.4 x 10⁴)
Since this is multiplication we can rearrange our factors. Recall the commutative property of multiplication allows us to perform this action:
(3.15 x 2.4)(10³ x 10⁴)
3.15 x 2.4 = 7.56
10³ x 10⁴ = 10^{3 + 4} = 10⁷
(3.15 x 10³)(2.4 x 10⁴) = 7.56 x 10⁷
We will move our decimal point seven places to the right:
75,600,000
(3.15 x 10³)(2.4 x 10⁴) = 75,600,000
Example 6: Find each quotient. $$\frac{2.97 \cdot 10^2}{9 \cdot 10^{-3}}$$ We can use our quotient rule for exponents to simplify the powers of 10: $$\frac{2.97 \cdot 10^2}{9 \cdot 10^{-3}}=\frac{2.97 \cdot 10^{2 - (-3)}}{9}$$ $$\frac{2.97 \cdot 10^{2 - (-3)}}{9}=\frac{2.97}{9}\cdot 10^5$$ $$\frac{2.97}{9}\cdot 10^5=0.33 \cdot 10^5$$ We will move our decimal point five places to the right: $$33{,}000$$

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