Calculations with Exponential Numbers

The objectives of this module are to learn to:

    1.    add exponential numbers
    2.    subtract exponential numbers
    3.    multiply exponential numbers
    4.    divide exponential numbers

I.    Introduction

In this module, you will learn how to perform the basic arithmetic processes of addition, subtraction, multiplication and division on numbers which have been converted to exponential notation.  First we will review the conversion of numbers to exponential notation.  The number 5678 is shown here in six different forms:

5.678 x 10 3       56.78 x 10 2   567.8 x 10 1
0.5678 x 10 4     0.05678 x 10 5   0.005678 x 10 6

As the decimal point is moved to the right, the value of the exponent decreases by one for each move.  As the decimal point is moved to the left, the value of the exponent increases by one for each move. However,......

The value of all of these numbers is exactly the same.

Exponential numbers can be changed to a form which has a higher value for the exponent without changing the magnitude of the expression.  1.234x10 4 can be changed to 0.1234x10 5 by moving the decimal place to the left one place and adding one to the exponent.  6.543x10 -3 has a negative exponent.  It can be changed to 0.6543x10 -2 by the same process.  (Remember that this is a larger exponent, since 10 -2 is greater than 10 -3.)

In the same way, exponential numbers can be changed to a form with a lower exponent.  2.468x10 5 can be changed to 24.68x10 4 by moving the decimal one place to the right and subtracting one from the exponent.   However, the standard exponential form of any number is

5.678x10 3

In this form, the coefficient (pre-exponential) number lies between 1 and 10.

    Do the following practice problems and check the solutions at the end of this module.


The following numbers are not written in standard exponential notation. Rewrite them so that they are in standard exponential notation.

Number Answer:
350 x 10 -5 3.5x10 -3 
0.0458 x 106 4.58x104   
0.0028x10 -3  2.8 x 10 -6

If your answers are not correct and if you do not thoroughly understand how to convert numbers to standard exponential notation, review the module Writing Numbers in Exponential Notation.

III.    Addition and Subtraction

The rules for addition and subtraction of exponents are the same, so they will be considered together.  To add two exponential numbers, you must first convert the exponents to the same value.  It is often easier to change the exponent of the smaller number to that of the larger number.

To solve addition problems with exponential numbers:

Several examples follow that illustrate the process for solving addition and subtraction problems.


Sample Problem:   What is 4.0x10 -2 + 5.0x10 -3 ?

Solution:   First, rewrite the smaller number so that it has the same exponent as that of the larger number:  This yields:
5.0x10 -3 = 0.50x10 -2

Then, add the coefficients: 4.0 + 0.5 = 4.5

Finally, multiply by the common exponent: 4.5x10 -2

The rules for subtraction are the same as the rules of addition.   To subtract two exponential numbers, you must also convert them to the same exponent by changing the value of the lower exponent into that of the higher.


Sample Problem:   Subtract 2.0 x 10 -7 from 5.8 x 10 -5.

Solution:   Rewrite the smaller number so that it has the same exponent as that of the larger number:  2.0 x 10 -7 would be rewritten as 0.02 x 10 -5

Subtract the coefficients: 5.8 - 0.020 = 5.78

Multiply by the common exponent:  5.78 x 10 -5

We would round this number to 5.8x10 -5 because the rules for significant figures for addition and subtraction only allow the final sum to have the same number of significant digits to the right of the decimal point as the number with the least significant digits after the decimal in the original addition or subtraction.   This can be easily seen if the numbers are written in normal form:

  0.000058
-0.00000020
  0.0000578    or   5.78x10 -5

You can use the step where you add or subtract the coefficients to determine the number of significant figures in your answer.  One final example of addition and subtraction of exponential numbers:


Sample Problem:   Add (3.15 x 10 -1) and (-2.05 x 10 1).

Solution:   Convert the smaller exponent to that of the larger: 3.15 x 10 -1 = 0.0315 x 10 1

Then add the coefficients, paying attention to significant figures:   -2.05 + 0.0315 = -2.0185 = -2.02

Finally, multiply by the common exponent: -2.02x10 1

Do the following problems:

Number Answer:
(6.04 x 10 3) + (2.60 x 10 2) = ? 6.30 x 103
(9.82 x 10 -4) - (8.20 x 10 -5) = ? 9.00 x 10-4
(4.3 x 10 -2) - (-1.2 x 10 -3) = ? 4.4 x 10-2

III.    Multiplication and Division

One of the main advantages of exponential notation is that the process of multiplication and division is greatly simplified.  Let's take a simple example, namely the multiplication of 10 2 · 10 3.

10 2 is 10 · 10
10 3 is 10 · 10 · 10
Thus, 10 2 · 10 3 = 10 (2+3) or 10 5


Sample Problem:   Multiply 10 -4 · 10 -7 · 10 -2.

Solution:   The simple rule for multiplying exponential numbers is to add the exponents.  Therefore,

10 -4 · 10 -7 · 10 -2 = 10 (-4 -7 -2) = 10 -13

Multiplying more complex exponential numbers requires two steps:
                Step 1:    Multiplication of the coefficients.
                Step 2:    Addition of the exponents.

For example, multiply (3 x 10 -2) · (2 x 10 -4).

First, multiply the coefficients: 3 · 2 = 6; 
Next, add the exponents: 10 -2 · 10 -4 = 10 (-2 + (-4)) = 10 -6
Combining the terms, gives 6 x 10 -6.


Sample Problem:   Multiply (3.1x10 -3) · (2.2x10 5).

Solution:  

Multiply the coefficients: 3.1 · 2.2 = 6.8
Add the exponents: 10 -3 + 10 5 = 10 (-3 +5) = 10 2

The answer is 6.8 x 102.

Division of exponential numbers is also a simple operation,

consisting of subtracting the exponents (5 - 2).  the difference is the value of the new exponent. An alternate approach when dividing exponential numbers is to recall that a negative exponent can be written as the reciprocal of that number.   For example, 10 5 ÷ 10 2 can be rewritten as a multiplication problem:

10 5 ÷ 10 2 = 10 5 · 1/10 2

Recall that 1/10 2 is the same thing as 10 -2.  Therefore, 10 5 ÷ 10 2 can be written as 10 5 · 10 -2 = 10 3.

Dividing more complex exponential numbers also requires also requires two steps:

Step 1:    Division of the coefficients.
Step 2:    Subtraction of the exponents.

For example, divide (6.01 x 10 -3) by (5.23 x 10 -6).

Step 1:    Divide the coefficients:    6.01 ÷ 5.23 = 1.15
Step 2:    Subtract the exponents:   10 -3 ÷ 10 -6 = 10 (-3 - (-6)) = 10 3

The answer is 1.15x10 3.

Do the following problems:

Number Answer:
(5.00 x 10 4) · (1.60 x 10 -2) = ? 8.00 x 102
(5.8 x 10 -2) ÷ (8.1 x 10 3) = ? 7.2 x 10-6 (always report your answer in standard exponential notation)

Let's summarize the rules for calculations with exponential:

Addition and subtraction:

1. Convert the exponents to the same value. To do this, change the exponent of the smaller number to that of the larger number.
2. Add or subtract the coefficients.
3. Multiply the result by the common exponent.

Multiplication and division:

1. Multiply or divide the coefficients.
2. For multiplication, add the exponents.  For division, subtract the exponents.

Of course, most of you will be do exponents on your calculators, but you should understand how to do them by hand. Be sure you know how to enter numbers with positive and negative exponents. Review the instruction booklet that came with the calculator, or ask the Science Learning Center staff for help. You can practice for the post test in the Practice Questions. When you feel confident that you understand the material, make your second attempt at the Math Skills Assessment.

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