The form is strict.
a × 10n, where 1 ≤ a < 10
correct
4.5 × 10⁴
not standard
45 × 10³
a must be under 10
also not
0.45 × 10⁵
a must be at least 1
All three are the same number. Only the first is written properly.
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Scientific Notation
One digit before the point, and a power of ten carrying the size.
Calculate
(3 × 105)(2 × 103)
Multiply the front numbers and add the powers: 3 × 2 = 6, and 10⁵ × 10³ = 10⁸. So 6 × 10⁸. D multiplies the exponents — powers add under multiplication, they never multiply.
Count the hops.
45,000 → 4.5
the point moves 4 left
= 4.5 × 10⁴
big number, positive power
0.00032 → 3.2
the point moves 4 right
= 3.2 × 10⁻⁴
small number, negative power
The sign tells you the size
A positive exponent means a number bigger than 1; a negative one means smaller than 1. If your sign disagrees with the original, the point moved the wrong way.
Multiplying and dividing.
Handle the numbers and the powers separately.
(3 × 10⁵)(2 × 10³)
the question
3 × 2 = 6, and 10⁵ ⁺ ³ = 10⁸
multiply, then add the powers
= 6 × 10⁸
already in standard form
(8 × 10⁷) ÷ (2 × 10³)
now dividing
= 4 × 10⁴
8 ÷ 2, and subtract the powers
The same split gives a quick size estimate. Round each number to one digit first: 2,960 × 0.0412 ≈ (3 × 10³)(4 × 10⁻²) = 12 × 10¹ = 120.
Tidying an answer that drifts out of range.
(6 × 10⁴)(5 × 10³)
6 × 5 = 30
= 30 × 10⁷
30 is too big
30 = 3.0 × 10¹
one more hop
= 3 × 10⁸
the exponent goes up by 1
Move the point left, the exponent grows. Move it right, the exponent shrinks. The number itself never changes.
Adding needs matching exponents.
Unlike multiplying, you cannot just work on the parts.
3 × 10⁵ + 4 × 10⁴
different powers
4 × 10⁴ = 0.4 × 10⁵
rewrite to match
(3 + 0.4) × 10⁵
now they combine
= 3.4 × 10⁵
340,000
Never add the exponents when adding
Powers add when you multiply. For a sum they have to be made equal first — it is the same idea as a common denominator.
Variations: a front number below 1, negative-power tidying, and a square root.
The front number falls below 1: (1.2 × 10⁻³) ÷ (4 × 10⁵).
1.2 ÷ 4 = 0.3, and −3 − 5 = −8
numbers and powers apart
0.3 × 10⁻⁸ = 3 × 10⁻¹ × 10⁻⁸
0.3 is too small
= 3 × 10⁻⁹
point right one, exponent down one
What is different: the tidy goes the other way. A front number under 1 pushes the exponent down, not up.
Tidying with a negative exponent: 52 × 10⁻⁶ and 0.045 × 10⁻³.
52 = 5.2 × 10¹, so 52 × 10⁻⁶ = 5.2 × 10⁻⁵
−6 + 1 = −5
0.045 = 4.5 × 10⁻², so 0.045 × 10⁻³ = 4.5 × 10⁻⁵
−3 + (−2) = −5
What is different: rewrite the front number as its own power of ten and add. Guessing the direction on a negative exponent gives 5.2 × 10⁻⁷, which is 100 times too small.
A square root: √1.6 × 10⁷.
1.6 × 10⁷ = 16 × 10⁶
make the exponent even first
√16 × √10⁶ = 4 × 10³
halve the exponent
= 4 × 10³ = 4,000
check: 4,000² = 16,000,000 ✓
What is different: an odd exponent cannot be halved, so we borrow a power of ten into the front number before taking the root.
Negative exponents, powers and “how many fit”.
The same laws, with the signs handled carefully. Dividing subtracts the exponents — even when they are negative.
(6.3 × 10⁻⁴) ÷ (2.1 × 10⁻⁹)
numbers and powers apart
6.3 ÷ 2.1 = 3, and −4 − (−9) = +5
minus a negative is a plus
= 3 × 10⁵
a tiny number ÷ a tinier one is big
(7 × 10⁻²)² = 49 × 10⁻⁴ = 4.9 × 10⁻³
square the 7, double the exponent, tidy
A grain is 4 × 10⁻⁴ m long. How many grains, end to end, make 2 cm?
2 cm = 2 × 10⁻² m
same units first
(2 × 10⁻²) ÷ (4 × 10⁻⁴) = 0.5 × 10²
−2 − (−4) = 2
= 50 grains = 5 × 10¹
tidy: 0.5 → 5, exponent down one
Three sign slips
Adding the exponents when dividing gives 10⁻¹³ instead of 10⁵. Squaring the exponent gives 10⁴ instead of 10⁻⁴ — a power of a power multiplies: −2 × 2 = −4. And leaving 2 cm as 2 m makes the count 100 times too big.
On the exam
Scientific-notation items mix the four operations with negative exponents and a final tidy into 1 ≤ a < 10, often inside a science story. Work the numbers and the powers of ten separately, then check the size: dividing by something tiny must give something big.
Check: write them out in full.
4.5 × 10⁴
45,000
✓
3.2 × 10⁻⁴
0.00032
✓
6 × 10⁸
300,000 × 2,000
600,000,000 ✓
4 × 10⁴
80,000,000 ÷ 2,000
40,000 ✓
Now you try
Practice problem
Calculate
5 × 106 + 8 × 105
Addition needs the powers to match first: 8 × 10⁵ = 0.8 × 10⁶, so the total is 5.8 × 10⁶. A adds the exponents as if multiplying. Multiplying never needs a common power; adding always does.