8.4: Scientific Notation
Sometimes in mathematics numbers are huge. They are so huge that we use what is called scientific notation. It is easier to work with such numbers when we shorten their decimal places and multiply them by 10 to a specific power. In this lesson, you will learn how to use scientific notation by hand and on a calculator.
Powers of 10:
10,000 &= 10^4\\
1,000 &= 10^3\\
100 &= 10^2\\
10 &= 10^1
Using Scientific Notation for Large Numbers
Example: If we divide 643,297 by 100,000 we get 6.43297. If we multiply this quotient by 100,000, we get back to our original number. But we have just seen that 100,000 is the same as
Solution: Look at the following examples:
2.08 \times 10^3 &= 2,080\\
2.08 \times 10^2 &= 208\\
2.08 \times 10^1 &= 20.8\\
2.08 \times 10^0 &= 2.08
The power tells how many decimal places to move; positive powers mean the decimal moves to the right. A positive 4 means the decimal moves four positions the right.
Example 1: Write in scientific notation.
653,937,000
Solution:
Oftentimes we do not keep more than a few decimal places when using scientific notation and we round the number to the nearest whole number, tenth, or hundredth depending on what the directions say. Rounding Example 1 could look like
Using Scientific Notation for Small Numbers
We’ve seen that scientific notation is useful when dealing with large numbers. It is also good to use when dealing with extremely small numbers.
Look at the following examples:
2.08 \times 10^{2} &= 0.0208\\
2.08 \times 10^{3} &= 0.00208\\
2.08 \times 10^{4} &= 0.000208
Example 2: The time taken for a light beam to cross a football pitch is 0.0000004 seconds. Write in scientific notation.
Solution:
Evaluating Expressions Using Scientific Notation
When evaluating expressions with scientific notation, it is easiest to keep the powers of 10 together and deal with them separately.
Example:
Solution: It is best to keep one number before the decimal point. In order to do that, we had to make 27.84 become
Example 3: Evaluate the following expression.
(a)
(b)
Solution:
(a)
(b)
You must remember to keep the powers of ten together, and have 1 number before the decimal.
Scientific Notation Using a Calculator
Scientific and graphing calculators make scientific notation easier. To compute scientific notation, use the [EE] button. This is [2nd] [,] on some TI models or
For example to enter
When you hit [ENTER] the calculator displays
Solving RealWorld Problems Using Scientific Notation
Example: The mass of a single lithium atom is approximately one percent of one millionth of one billionth of one billionth of one kilogram. Express this mass in scientific notation.
Solution: We know that percent means we divide by 100, and so our calculation for the mass (in kg) is
Next, we use the product of powers rule we learned earlier in the chapter.
The mass of one lithium atom is approximately
Practice Set
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both.
CK12 Basic Algebra: Scientific Notation (14:26)
Write the numerical value of the following expressions.

3.102×102 
7.4×104 
1.75×10−3 
2.9×10−5 
9.99×10−9 
(3.2×106)⋅(8.7×1011) 
(5.2×10−4)⋅(3.8×10−19) 
(1.7×106)⋅(2.7×10−11) 
(3.2×106)÷(8.7×1011) 
(5.2×10−4)÷(3.8×10−19) 
(1.7×106)÷(2.7×10−11)
Write the following numbers in scientific notation.
 120,000
 1,765,244
 63
 9,654
 653,937,000
 1,000,000,006
 12
 0.00281
 0.000000027
 0.003
 0.000056
 0.00005007
 0.00000000000954
 The moon is approximately a sphere with radius
r=1.08×103 miles. Use the formula SurfaceArea=4πr2 to determine the surface area of the moon, in square miles. Express your answer in scientific notation, rounded to two significant figures.  The charge on one electron is approximately
1.60×10−19 coulombs. One Faraday is equal to the total charge on6.02×1023 electrons. What, in coulombs, is the charge on one Faraday?  Proxima Centauri, the next closest star to our Sun, is approximately
2.5×1013 miles away. If light from Proxima Centauri takes3.7×104 hours to reach us from there, calculate the speed of light in miles per hour. Express your answer in scientific notation, rounded to two significant figures.
Mixed Review
 14 milliliters of a 40% sugar solution was mixed with 4 milliliters of pure water. What is the concentration of the mixture?
 Solve the system \begin{cases} 6x+3y+18\\ 15=11y5x \end{cases}.
 Graph the function by creating a table: @$f(x)=2x^2@$. Use the following values for @$x: 5 \le x \le 5@$.
 Simplify @$\frac{5a^6 b^2 c^{6}}{a^{11} b}@$. Your answer should have only positive exponents.
 Each year Americans produce about 230 million tons of trash (Source: http://www.learner.org/interactives/garbage/solidwaste.html). There are 307,006,550 people in the United States. How much trash is produced per person per year?
 The volume of a 3dimesional box is given by the formula: @$V=l(w)(h)@$, where @$l=@$ length, @$w=@$ width, and @$h=@$ height of the box. The box holds 312 cubic inches and has a length of 12 inches and a width of 8 inches. How tall is the box?
Quick Quiz
 Simplify: @$\frac{(2x^{4}y^3)^{3} \ \cdot \ x^{3} y^{2}}{2x^0y^2}@$.
 The formula @$A=1,500(1.0025)^t@$ gives the total amount of money in a bank account with a balance of $1,500.00, earning 0.25% interest, compounded annually. How much money would be in the account five years in the past?
 True or false? @$\left(\frac{5}{4}\right)^{3}= \frac{125}{64}@$