Operations with Scientific Notation — Print
Handle the front numbers and the powers of ten separately, then tidy the result back into proper form.
What the practice questions cover
These questions use 4 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Rate — A ratio of two different units, like miles per hour or dollars per pound.
- Exponent — How many times the base is used as a factor. In 53 the exponent is 3.
- Scientific notation — A number written as a × 10n, where 1 ≤ a < 10.
- Standard form — The ordinary way of writing a number, like 47,000.
Light travels 300 000 000 metres every second.
Read that number again and say how many zeros it had. Most people have to go back and count.
Now write it as 3 × 108. The 8 does the counting for you, and an 8 cannot slip the way a row of zeros can.
It says: start at 3 and slide the decimal point eight places right.
The same trick runs the other way. A virus is 0.00000012 m across — or 1.2 × 10-7, where the minus means slide left.
This is why every number in science is written this way. Not to look clever, but because comparing two exponents is easy and comparing two rows of zeros is not.
The form itself
The front number must be at least 1 and less than 10. So 4.7 × 104 is correct; 47 × 103 is not, even though both equal 47,000.
A positive exponent means a big number. A negative exponent means a small one, between 0 and 1.
Converting: count how many places the decimal point has to move. Left moves make the exponent go up.
Negative exponent does not mean negative number
3 × 10-5 is 0.00003 — small, but positive. Only a minus in front of the 3 would make it negative.
Multiplying and dividing
Multiply the front numbers; add the exponents. Divide the front numbers; subtract the exponents.
(3 × 105)(2 × 107) = 6 × 1012.
If the new front number falls outside 1 to 10, fix it: 20 × 104 becomes 2 × 105.
Always tidy at the end
A front number of 20 or of 0.5 means you are not finished. Move one factor of ten across and adjust the exponent.
Adding and subtracting
You may only add the front numbers when the exponents match. 3 × 106 + 4 × 106 = 7 × 106.
If they differ, rewrite one so they agree. 3 × 106 + 4 × 105 = 3 × 106 + 0.4 × 106 = 3.4 × 106.
This is the same logic as needing a common denominator before adding fractions.
Match, add, tidy
Three steps in that order. Trying to add mismatched exponents gives answers that are wrong by factors of ten.
Comparing sizes
Compare the exponents first. A larger exponent wins outright, whatever the front numbers are.
Only when the exponents are equal do you compare the front numbers.
"How many times bigger?" means divide: 6 × 1082 × 105 = 3 × 103, so three thousand times.
Worked examples
Multiplying
Compute (5 × 106)(4 × 10-3) in scientific notation.
- Front numbers: 5 × 4 = 20 Multiply the coefficients.
- Powers: 106 · 10-3 = 103 Add the exponents: 6 + (-3) = 3.
- So far 20 × 103, but 20 is too big. The front must sit between 1 and 10.
- 20 = 2 × 101, so the answer is 2 × 104 One more factor of ten moves across.
2 × 104
Adding with different exponents
Compute 7.2 × 105 + 3 × 104.
- The exponents differ, so rewrite the smaller one. You cannot add until they match.
- 3 × 104 = 0.3 × 105 Move the decimal left one place, raise the exponent by one.
- 7.2 × 105 + 0.3 × 105 = 7.5 × 105 Now add the front numbers.
- 7.5 is between 1 and 10, so it is finished. No tidying needed.
7.5 × 105
How many times bigger?
The Sun is about 1.4 × 106 km across; the Earth about 1.3 × 104 km. How many times wider is the Sun?
- Divide: 1.4 × 1061.3 × 104 "How many times" means divide.
- Front numbers: 1.4 ÷ 1.3 ≈ 1.08 Just over one.
- Powers: 106 ÷ 104 = 102 Subtract the exponents.
- ≈ 1.08 × 102 ≈ 108 times About a hundred times wider.
About 108 times
Words to know
- Scientific notation
- a × 10n where 1 ≤ a < 10 and n is an integer.
- Order of magnitude
- The power of ten — roughly how many digits the number has.
- Coefficient
- The front number a.
- Standard form
- The ordinary way of writing a number, like 47,000.
Mistakes to watch for
Leaving the front number outside 1 to 10. 47 × 103 has the right value but is not scientific notation — and two numbers written that way cannot be compared by exponent, which was the entire purpose.
Adding the front numbers while the exponents still differ. 3 × 106 + 4 × 105 is not 7 × 106. Rewrite one so they match first — it is the same reason fractions need a common denominator.
Multiplying the exponents instead of adding them. 103 × 104 = 107, because you are stacking three tens next to four tens. Multiplying gives 1012, which is out by a factor of a hundred thousand.
Reading a negative exponent as a negative number. 3 × 10-5 is 0.00003 — small, but firmly positive. Only a minus in front of the 3 would make the number itself negative.
Archimedes wrote The Sand Reckoner around 250 BC to answer a taunt: that the number of grains of sand was beyond counting.
He invented a notation for enormous numbers, then estimated the grains needed to fill the universe as about 1063.
He had no zero, no decimal point and no exponent notation. He built the whole system from scratch just to make the point.
Problem 1
Write 47,000 in scientific notation.
Worked answer
4.7 x 10^4
4.7, moved 4 places
4.7 × 104
Problem 2
Write 0.00062 in scientific notation.
Worked answer
6.2 x 10^-4
6.2, moved 4 places right
6.2 × 10-4
Problem 3
Write 3.5 × 105 in standard form.
Worked answer
350000
350,000
Problem 4
Compute (2 × 103)(4 × 105).
Worked answer
8 x 10^8
2 × 4 = 8
3 + 5 = 8
8 × 108
Problem 5
Compute 9 × 1073 × 102.
Worked answer
3 x 10^5
9 ÷ 3 = 3
7 - 2 = 5
3 × 105
Problem 6
Compute (5 × 104)(6 × 103) in proper scientific notation.
Worked answer
3 x 10^8
5 × 6 = 30 and 4 + 3 = 7
30 × 107 — front is too big
= 3 × 108
Problem 7
Compute (8 × 10-3)(2 × 106).
Worked answer
1.6 x 10^4
8 × 2 = 16
-3 + 6 = 3
16 × 103 = 1.6 × 104
Problem 8
Compute 4.5 × 106 + 2 × 106.
Worked answer
6.5 x 10^6
4.5 + 2 = 6.5
6.5 × 106
Problem 9
Compute 8 × 105 + 3 × 104.
Worked answer
8.3 x 10^5
3 × 104 = 0.3 × 105
8 + 0.3 = 8.3
8.3 × 105
Problem 10
Which is larger, 3 × 108 or 9 × 107?
- 3 × 108
- 9 × 107
- They are equal
- Cannot be determined
Worked answer
3 × 108
108 > 107
3 × 108 = 300,000,000
9 × 107 = 90,000,000
Problem 11
How many times larger is 6 × 109 than 2 × 106?
Worked answer
3000
6 ÷ 2 = 3
109 ÷ 106 = 103
3 × 103 = 3000
Problem 12
Rewrite 0.045 × 107 in proper scientific notation.
Worked answer
4.5 x 10^5
0.045 = 4.5 × 10-2
-2 + 7 = 5
4.5 × 105
Problem 13
Compute 7.2 × 108 - 4 × 107.
Worked answer
6.8 x 10^8
4 × 107 = 0.4 × 108
7.2 - 0.4 = 6.8
6.8 × 108
Problem 14
A virus is 1.2 × 10-7 m across; a cell is 3 × 10-5 m. How many times wider is the cell?
Worked answer
250
3 ÷ 1.2 = 2.5
10-5 ÷ 10-7 = 102
2.5 × 102 = 250 times
Problem 15
Compute 2 × 1048 × 10-2 in scientific notation.
Worked answer
2.5 x 10^5
2 ÷ 8 = 0.25
4 - (-2) = 6
0.25 × 106 = 2.5 × 105
Problem 16
Light travels 3 × 108 m/s. How far does it travel in 2 × 103 seconds?
Worked answer
6 x 10^11
3 × 2 = 6
8 + 3 = 11
6 × 1011 metres