INTERMEDIATE ALGEBRA

SUMMER TERM 2001

Naturals, Integers, Fractions, Decimals, Rationals, Square Roots, Irrationals, Reals, Formulas, Graphing

**Scientific Notation** was developed to easily represent
numbers that are either very large or very small. Here are two examples of large and small
numbers. They are expressed in decimal form instead of scientific notation to help
illustrate the problem:

- The mass of the Earth is 5,980,000,000,000,000,000,000,000 kilograms.

- The mass of an alpha particle, which is emitted in the radioactive decay of Plutonium-239, is 0.000,000,000,000,000,000,000,000,006,645 kilograms.

As you can see, it could get tedious writing out those zeros repeatedly, so **Scientific
Notation** was developed to help represent these numbers in a way that was easy to
read and understand.

**Large Numbers** - The mass of the Earth can be written as -

m_{E} = 5.98 x 1,000,000,000,000,000,000,000,000 kg

It is that large number, 1,000,000,000,000,000,000,000,000 which cause the problem. But that is just a multiple of ten. In fact it is ten times itself twenty four times. A more convenient way of writing it is

1,000,000,000,000,000,000,000,000 = 10^{24}.

The small number to the right of the ten is called the "exponent," or the "power of ten." It represents the number of zeros that follow the 1. This enables us to write the mass of the Earth as -

m_{E} = 5.98 x 10^{24} kg

**Small Numbers** - The mass of an alpha particle can be written as -

m_{a} = 6.645 x
0.000,000,000,000,000,000,000,000,001 kg

It is that small number, 0.000,000,000,000,000,000,000,000,001 which cause the problem. But that is just a multiple of one tenth. In fact it is one tenth times itself twenty seven times. A more convenient way of writing it is

0.000,000,000,000,000,000,000,000,001 = (1/10)^{27} = 10^{-27}.

Again the small number to the right of the ten is called the "exponent," or the "power of ten." Note that here it is negative, and it represents the decimal position of the 1. This enables us to write the mass of the alpha particle as -

m_{a} = 6.645 x 10^{-27} kg

**Definition**: Any decimal number can be written as a product of a
decimal number between 1 and 10 multiplied by an integer power of ten. When this is done
the number is written in **scientific notation**.

**Complete the following problems on a separate sheet of paper. Be sure to show
all of the work necessary to complete the problems**. (Calculators may be used to
check the answer, but should not be used to solve the problem originally.)

**Numbers Larger than 10**

* Example*:

1. Write 670,620 in scientific notation.

Step 1 - Move the decimal point of 670,620 to the left to get a number between 1 and 10 times a power of ten.

670,620 = 6.7062 x 10^{exponent}

Step 2 - Find the exponent by counting how many places you had to move the decimal point. This is the value of the exponent.

exponent = 5

Step 3 - Write the number in scientific notation.

670,620 = 6.7062 x 10^{5}

**HOMEWORK PROBLEMS** :

2. Write 340.67 in scientific notation.

3. Write 93,000,000 in scientific notation.

4. Write 300,000,000 in scientific notation.

5. Write 1,840,000,000,000,000 in scientific notation.

**Numbers smaller than 1 **

* Example*:

6. Write 0.00234 in scientific notation.

Step 1 - Move the decimal point of 0.00234 to the right to get a number between 1 and 10 times a power of ten.

0.00234 = 2.34 x 10^{exponent}

Step 2 - Find the exponent by counting how many places you had to move the decimal point. This is the value of the exponent.

exponent = -3

Step 3 - Write the number in scientific notation.

0.00234 = 2.34 x 10^{-3}

**HOMEWORK PROBLEMS** :

7. Write 0.00005723 in scientific notation.

8. Write 0.00000001032 in scientific notation.

9. Write 0.000000000025789 in scientific notation.

10. Write 0.0000000000000000000097531 in scientific notation.

**Addition** - The key to adding numbers in Scientific Notation is to make
sure the exponents are the same.

* Example*:

11. Add 6.7062 x 10^{5} plus 3.4067 x 10^{2}.

Step 1 - Rewrite both numbers with the same power of 10.

6.7062 x 10^{5} = 6706.2 x 10^{2}

3.4067 x 10^{2} = 3.4067 x 10^{2}

Step 2 - Add the decimal numbers and multiply by the common power of 10.

6706.2 x 10^{2} + 3.4067 x 10^{2} = 6709.6067 x 10^{2}

Step 3 - Write the answer in scientific notation.

6.7062 x 10^{5} + 3.4067 x 10^{2} = 6.7096067 x 10^{5}

**HOMEWORK PROBLEMS** :

12. Add 3 x 10^{8} plus 9.3 x 10^{7}.

13. Add 9.3 x 10^{7} plus 6.7062 x 10^{5}.

14. Add 6.7845 x 10^{-6} plus 2.34 x 10^{-2}.

15. Add 8.205 x 10^{2} plus 3.4067 x 10^{-1}.

**Subtraction** - The key to subtracting numbers in Scientific Notation is
the same as for addition.

* Example*: 16. Subtract 6.7062 x 10

Step 1 - Rewrite both numbers with the same power of 10.

6.7062 x 10^{4} = 6.7062 x 10^{4}

3.4067 x 10^{5} = 34.067 x 10^{4}

Step 2 - Subtract the decimal numbers and multiply by the common power of 10.

34.067 x 10^{4} - 6.7062 x 10^{4} = 27.3608 x 10^{4}

Step 3 - Write the answer in scientific notation.

3.4067 x 10^{5} - 6.7062 x 10^{4} = 2.73608 x 10^{5}

**HOMEWORK PROBLEMS** :

17. Subtract 3 x 10^{8} from 9.3 x 10^{7}.

18. Subtract 9.3 x 10^{7} from 6.7062 x 10^{5}.

19. Subtract 6.7845 x 10^{-6} from 3.4067 x 10^{-5}.

20. Subtract 6.7062 x 10^{-2} from 3.4067 x 10^{-1}.

**Multiplication** - When multiplying numbers expressed in Scientific
Notation, the exponents can simply be added together.

* Example*:

21. Multiply 6.7062 x 10^{5} times 3.4067 x 10^{2}.

Step 1 - Multiply the decimal numbers together and multiply the powers of 10 together.

6.7062 x 10^{5} * 3.4067 x 10^{2} = 6.7062*3.4067 x 10^{5}
x 10^{2}

Step 2 - Express the product of the decimal numbers in scientific notation.

6.7062*3.4067 x 10^{5} x 10^{2} = 2.284601154 x 10^{1}
x 10^{5} x 10^{2}

Step 3 - Write the answer in scientific notation by adding up all of the exponents.

6.7062 x 10^{5} * 3.4067 x 10^{2}= 2.284601154 x 10^{8}

**HOMEWORK PROBLEMS** :

22. Multiply 6.2 x 10^{4} times 3.40 x 10^{7}.

23. Multiply 9.3 x 10^{7} times 2.34 x 10^{2}.

24. Multiply 6.843 x 10^{-3} times 3.4067 x 10^{2}.

25. Multiply 5.932 x 10^{-6} times 8.63 x 10^{-3}.

**Division** - When dividing numbers expressed in Scientific Notation, the
exponent of the denominator is subtracted from the exponent of the numerator.

* Example*:

26. Divide 6.7062 x 10^{5} by 3.4067 x 10^{2}.

Step 1 - Divide the decimal numbers together and divide the powers of 10 together.

(6.7062 x 10^{5} ) / (3.4067 x 10^{2} )= (6.7062 /
3.4067) x (10^{5} / 10^{2} )

Step 2 - Express the quotient of the decimal numbers in scientific notation and subtract the denominator exponent from the numerator exponent.

(6.7062/3.4067) x (10^{5} / 10^{2} ) = 1.968532598 x 10^{0}
x 10^{3}

Step 3 - Write the answer in scientific notation by adding up all of the remaining exponents.

(6.7062 x 10^{5} ) /( 3.4067 x 10^{2} )= 1.968532598 x
10^{3}

**HOMEWORK PROBLEMS** :

27. Divide 6.2 x 10^{4} by 3.40 x 10^{7}.

28. Divide 9.3 x 10^{7} by 2.34 x 10^{2}.

29. Divide 6.843 x 10^{-3} by 3.4067 x 10^{2}.

30. Divide 5.932 x 10^{-6} by 8.63 x 10^{-3}.

**Square Root** - When finding the square root of a number expressed in
Scientific Notation, it is necessary to write the number with an even power of 10, since
the exponent of the square root of a number is one-half the exponent of the original
number.

* Example*:

31. Find the square root of 6.7062 x 10^{5}.

Step 1 - If necessary, rewrite the number with an even power of 10 by moving the decimal one place to the right, and decreasing the exponent by one.

6.7062 x 10^{5} = 67.062 10^{4}

Step 2 - Find the square root of the decimal number and divide the exponent by 2 to get the answer.

**HOMEWORK PROBLEMS** :

32. Find the square root of 3.40 x 10^{7}.

33. Find the square root of 2.34 x 10^{12}.

34. Find the square root of 6.843 x 10^{-3}.

35. Find the square root of 5.932 x 10^{-6}.