One digit before the point, and an exponent that counts how far it moved.
Write in scientific notation
0.00047
Move the point right until one non-zero digit is in front: 0.00047 → 4.7, which took 4 places. Because the number is smaller than 1, the exponent is negative: 4.7 × 10⁻⁴. A has the sign backwards — a positive exponent would make it 47,000. C is arithmetically equal but not in proper form, since 47 is not between 1 and 10. D miscounts by one.
a × 10ⁿ, where 1 ≤ a < 10 and n is a whole number.
| Written | Proper? | Why |
|---|---|---|
| 3.6 × 10⁵ | yes | one digit before the point |
| 36 × 10⁴ | no | 36 is not below 10 |
| 0.36 × 10⁶ | no | 0.36 is below 1 |
| 1 × 10⁻³ | yes | 1 counts |
All four of those are the same size in pairs — being improper does not make a number wrong, only badly written. Exam marks usually depend on proper form.
4.7 × 10⁻⁴ is 0.00047, which is small but perfectly positive. A negative number would be −4.7 × 10⁴.
6.02 × 10⁴
positive → move right
= 60,200
four places, filling with zeros
3.1 × 10⁻⁵
negative → move left
= 0.000031
five places
Count the places moved, not the zeros written. 6.02 × 10⁴ has only two zeros in it, because two of the four places were already occupied by the 0 and the 2.
The last one catches people out: 9.9 looks larger than 5.0, but the exponent decides first, and −4 is a smaller exponent than −2.
To add or subtract, rewrite one number so both carry the same power of 10.
Simplify (7.2 × 10⁵) + (4 × 10⁴), in scientific notation.
4 × 10⁴ = 0.4 × 10⁵
raise the smaller one's exponent to match
7.2 × 10⁵ + 0.4 × 10⁵ = 7.6 × 10⁵
now add the fronts; the power stays
Check in standard form: 720,000 + 40,000 = 760,000 = 7.6 × 10⁵ ✓
Writing 7.2 + 4 = 11.2 and 5 + 4 = 9 gives 11.2 × 10⁹, which is over ten thousand times too big. Exponents are added only when multiplying; for a sum, the powers must first be made the same.
A story: one sheet of bond paper is about 1 × 10⁻¹ mm thick. How tall, in metres, is a stack of 2.5 × 10⁴ sheets?
(2.5 × 10⁴)(1 × 10⁻¹) = 2.5 × 10³ mm
multiply fronts, add exponents: 4 + (−1) = 3
2,500 mm ÷ 1,000 = 2.5 m
the question asked for metres
The scientific notation gives the size; the units still have to be converted at the end, exactly as in any measurement problem.
The same rules; these are the answers that come out untidy and need one more move.
Simplify (2.4 × 10³) ÷ (6 × 10⁵).
2.4 ÷ 6 = 0.4 and 3 − 5 = −2
divide fronts, subtract exponents
0.4 × 10⁻² = 4 × 10⁻³
point moves right, exponent goes down by one
Tidying a front number that is too big raises the exponent; one that is below 1 lowers it.
Simplify (5 × 10³)².
(5 × 10³)(5 × 10³)
squaring means two copies
= 25 × 10⁶
5 × 5 and 3 + 3
= 2.5 × 10⁷
tidy: 25 is not below 10
Both parts are squared: the 5 becomes 25 and the exponent doubles. Squaring only the power gives 5 × 10⁶.
Simplify (6.1 × 10⁻³) − (9 × 10⁻⁴).
9 × 10⁻⁴ = 0.9 × 10⁻³
−4 is the smaller exponent; raise it to −3
6.1 × 10⁻³ − 0.9 × 10⁻³ = 5.2 × 10⁻³
check: 0.0061 − 0.0009 = 0.0052 ✓
With negative exponents, −3 is the larger power, so it is the one both numbers move to.
Multiply the fronts, add the exponents. Divide the fronts, subtract the exponents.
(3 × 10⁴)(5 × 10⁶)
multiply
= 15 × 10¹⁰
3 × 5, and 4 + 6
= 1.5 × 10¹¹
15 is not below 10, so fix it
(8 × 10⁵) ÷ (2 × 10⁻³)
divide
= 4 × 105 − (−3)
subtracting a negative
= 4 × 10⁸
already proper
15 × 10¹⁰ is not a wrong value, but it is not scientific notation. Moving the point one place left raises the exponent by one, giving 1.5 × 10¹¹.
Expect a conversion each way, a "which is largest", and a product or quotient of two numbers in scientific notation, sometimes inside a science or distance story. The fastest route is fronts with fronts and exponents with exponents, tidying last; the 10-second check is that the front number sits between 1 and 10 and that a small quantity has a negative exponent.
Simplify
(6 × 10⁻³)(4 × 10⁸)
Multiply the fronts: 6 × 4 = 24. Add the exponents: −3 + 8 = 5. That gives 24 × 10⁵, which is not proper form, so move the point one left and raise the exponent: 2.4 × 10⁶. A stops one step early, and B tidies the front number but forgets to adjust the exponent with it. C subtracts the exponents instead of adding.
Hint: when you shift the point left to tidy the front number, the exponent goes up — the number as a whole has not changed size.
