Operations with Scientific Notation — Challenge
Handle the front numbers and the powers of ten separately, then tidy the result back into proper form.
What these problems 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.
Problem 1
Write 4,500 in scientific notation.
- 4.5× 103
- 45× 104
- 4.5× 104
- 4.5× 102
- 4.5× 10-3
Worked answer
4.5× 103
Scientific notation needs a number at least 1 and less than 10, multiplied by a power of ten.
Moving the point in 4500 three places left gives 4.5, so the power is 103.
Note that 45× 102 has the same value but is not in scientific notation, because 45 is outside the required range.
Problem 2
Write 3.2× 104 as an ordinary number.
- 320
- 3,200
- 32,000
- 320,000
- 0.00032
Worked answer
32,000
A positive power of ten moves the decimal point to the right, here by four places.
3.2 becomes 32,000.
Problem 3
Write 0.00056 in scientific notation.
- 5.6× 104
- 56× 10-3
- 5.6× 10-4
- 5.6× 10-3
- 0.56× 10-4
Worked answer
5.6× 10-4
Move the point right until the number lies between 1 and 10: 0.00056 becomes 5.6 after four places.
Moving right means the exponent is negative, so the answer is 5.6× 10-4.
Counting only the zeros gives 10-3, which is the common slip.
Problem 4
What is (2× 103)(3× 102)?
- 6× 106
- 6× 105
- 5× 105
- 6× 101
- 23× 105
Worked answer
6× 105
Multiply the front numbers and the powers separately.
2· 3 = 6, and multiplying powers of ten adds the exponents: 103· 102 = 105.
So the answer is 6× 105. Multiplying the exponents instead would give 106.
Problem 5
What is 105?
- 10,000
- 100,000
- 1,000,000
- 50
- 500,000
Worked answer
100,000
A power of ten is a 1 followed by as many zeros as the exponent.
105 is 1 followed by five zeros, which is 100,000.
Problem 6
What is (8× 106)÷(2× 102)?
- 4× 103
- 4× 104
- 6× 104
- 4× 108
- 16× 104
Worked answer
4× 104
Divide the front numbers and the powers separately.
8÷ 2 = 4, and dividing powers of ten subtracts the exponents: 106÷ 102 = 104.
So the answer is 4× 104. Adding the exponents would wrongly give 108.
Problem 7
Which is the larger number, 3× 105 or 9× 104?
- 3× 105
- 9× 104
- They are equal
- It cannot be decided
- Both are less than 1
Worked answer
3× 105
Compare the powers of ten first, because they dominate.
105 is ten times 104, so 3× 105 = 300,000 while 9× 104 = 90,000.
The larger front number does not win when the power is smaller.
Problem 8
Write 7.1× 102 as an ordinary number.
- 71
- 710
- 7,100
- 0.071
- 7.1
Worked answer
710
The exponent 2 moves the decimal point two places to the right.
7.1 becomes 710.