💡 Why Scientific Notation?
Scientists and mathematicians use scientific notation to write numbers that are too large or too small to write out normally. On the GED, you need to convert between standard and scientific notation and divide in scientific notation — like the amoeba problem from the screenshot!
a × 10n
a = a number that is ≥ 1 and < 10 (one digit before the decimal)
n = an integer exponent (positive for large numbers, negative for small)
4.5 × 104 | 3.2 × 106 | 8.0 × 10−7
All have one digit (1–9) before the decimal point.
45 × 103 ❌ (45 is not between 1 and 10)
0.45 × 105 ❌ (0.45 is less than 1)
Positive exponent (104) → large number → move decimal RIGHT
Negative exponent (10−3) → small number → move decimal LEFT
Which is valid scientific notation? 42 × 103 or 4.2 × 104?
Move the decimal point to the LEFT until you have one digit before it. Count how many places you moved — that number becomes the positive exponent of 10.
The decimal point is at the end: 45,000.
4.5000 — moved 4 places to the left
The coefficient is 4.5, the exponent is 4 (positive — it's a large number)
45,000 = 4.5 × 104
Convert 8,500,000 to scientific notation. (How many places does the decimal move?)
Move the decimal point to the RIGHT until you have one non-zero digit before it. Count how many places you moved — that number becomes the negative exponent of 10.
The decimal is at position: 0.0045
4.5 — moved 3 places to the right
Moved right 3 places → exponent is −3
0.0045 = 4.5 × 10−3
Convert 0.00032 to scientific notation. (Move decimal right until you get 3.2 — how many places?)
✅ 4.5 × 104 | 🚫 45 × 103 (45 is too big!)
✅ 1.0 × 106 | 🚫 0.5 × 107 (0.5 is too small!)
45,000 = 4.5 × 104 (exponent = +4, positive)
The bigger the number, the larger the positive exponent.
0.0045 = 4.5 × 10−3 (exponent = −3, negative)
The smaller the number, the more negative the exponent.
| Standard Form | Decimal Moves | Direction | Scientific Notation |
|---|---|---|---|
| 45,000 | 4 places | ← LEFT | 4.5 × 104 |
| 3,200,000 | 6 places | ← LEFT | 3.2 × 106 |
| 0.0045 | 3 places | → RIGHT | 4.5 × 10−3 |
| 0.0000008 | 7 places | → RIGHT | 8 × 10−7 |
| 4.0 × 10−6 (amoeba!) | 6 places | → RIGHT | 0.000004 g |
🔑 Memory Trick
Large number → move decimal LEFT → positive exponent
Small number → move decimal RIGHT → negative exponent
Think: "Left = Positive, Right = Negative" — or remember that 104 = 10,000 (large) while 10−4 = 0.0001 (small).
Divide the coefficients separately, then subtract the exponents:
(a × 10ᵐ) ÷ (b × 10ⁿ) = (a ÷ b) × 10ᵐ⁻ⁿ
💡 Shortcut for this type of problem
To find how many fit in 1 gram, think: 1 ÷ 4.0 = 0.25, and 1 ÷ 10⁻⁶ = 10⁶.
So 0.25 × 10⁶ = 2.5 × 10⁵. That's 250,000 amoebas!
Enter 0.00032 (the exercise from your lesson) | 0.000004 (the amoeba mass) | 250000 (the answer to the amoeba problem!)