Standard form calculations
In a nutshell
Standard form is used as an easy way to write large or small numbers. There are rules for adding, subtracting, multiplying and dividing numbers that are written in standard form.
Writing a number in standard index form
A number is in standard index form if it is written in the form:
a×10b
Where the number a is either an integer or a decimal that is always between 1 and 10, and b is a whole number that may be negative or positive.
Write a large or small number in standard form
PROCEDURE
1.
| Move the decimal point until the number is between 1 and 10. Count how many spaces the decimal point was moved.
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2.
| This number between 1 and 10 is the value of a.
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3.
| The number of spaces the decimal point was moved is the value of b. If the decimal point moved to the left, then b is positive. If the decimal point was moved to the right, then b is negative.
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Example 1
What is 76000 in standard index form?
First, write 76000 as 76000.00 and move the decimal point until the number is between 1 and 10:
7.6000↶↶↶↶00, so a=7.6
The decimal point moved 4 spaces to the left, so b=4.
76000=7.6×104
Example 2
What is 0.0000815 in standard index form?
First, move the decimal point until the number is between 1 and 10:
000008.↷↷↷↷↷15, so a=8.15
The decimal point moved 5 spaces to the right, so b=−5.
0.0000815=8.15×10−5
Multiplying and dividing in standard index form
To multiply and divide two numbers in standard form, use the rules of indices. Multiply/divide the numbers as normal, and use rules of indices to multiply or divide the powers of 10. To multiply the powers of 10, add the powers and to divide powers of 10, subtract the powers.
Example 3
What is (1.5×108)×(9×10−3)?
Multiply 1.5 and 9:
1.5×9=13.5
Use rules of indices to multiply the powers of 10:
108×10−3=108+(−3)=105
Combine the answers to give:
(1.5×108)×(9×10−3)=13.5×105
Adjust the number so that it is written in standard form. a should be between 1 and 10. So change 13.5 to 1.35 and increase the power of 10 by 1.
(1.5×108)×(9×10−3)=1.35×106
Example 4
What is (8×105)÷(1.25×102)?
Divide 8 by 1.25:
8÷1.25=6.4
Use rules of indices to divide the powers of 10:
105÷102=103
Combine the answers to give:
(8×105)÷(1.25×102)=6.4×103
Adding and subtracting in standard index form
To add and subtract numbers in standard index form, make sure that the powers of 10 are the same first. Then, add/subtract the numbers and adjust the answer so that the number is between 1 and 10.
Example 5
What is (8×104)+(5×102)?
First, make the powers of 10 the same. The easiest way to do this is to write:
8×104=8×102×102=800×102
Now, add the numbers together:
(800×102)+(5×102)=(800+5)×102=805×102
Rewrite the number so that the answer is in standard form:
805×102=8.05×102×102=8.05×104
(8×104)+(5×102)=8.05×104