Scientific Notation: A way to write very large or very small numbers in a compact form

General Form: \(a \times 10^n\)

Where: \(1 \leq a < 10\) and \(n\) is an integer

Coefficient (a): A number between 1 and 10 (can equal 1 but less than 10)

Base: Always 10

Exponent (n): An integer (positive, negative, or zero)

Example: \(6.02 \times 10^{23}\) → Coefficient: 6.02, Base: 10, Exponent: 23

✓ Makes large numbers easier to read: \(5,800,000,000 = 5.8 \times 10^9\)

✓ Makes small numbers easier to read: \(0.0000042 = 4.2 \times 10^{-6}\)

✓ Used in science for very large/small measurements

✓ Simplifies calculations with extreme values

Step 1: Move the decimal point to the left until you have a number between 1 and 10

Step 2: Count the number of places you moved the decimal

Step 3: That number becomes the positive exponent of 10

Rule: Decimal moves LEFT → Exponent is POSITIVE

\(45,000 = 4.5 \times 10^4\) (moved decimal 4 places left)

\(7,200,000 = 7.2 \times 10^6\) (moved decimal 6 places left)

\(93,000,000 = 9.3 \times 10^7\) (moved decimal 7 places left)

Step 1: Move the decimal point to the right until you have a number between 1 and 10

Step 2: Count the number of places you moved the decimal

Step 3: That number becomes the negative exponent of 10

Rule: Decimal moves RIGHT → Exponent is NEGATIVE

\(0.0056 = 5.6 \times 10^{-3}\) (moved decimal 3 places right)

\(0.000082 = 8.2 \times 10^{-5}\) (moved decimal 5 places right)

\(0.00000007 = 7 \times 10^{-8}\) (moved decimal 8 places right)

Rule: Move decimal point to the RIGHT

Step 1: Look at the exponent number

Step 2: Move the decimal point that many places to the right

Step 3: Add zeros if needed

\(3.5 \times 10^3 = 3,500\) (move decimal 3 places right)

\(6.02 \times 10^5 = 602,000\) (move decimal 5 places right)

\(1.8 \times 10^4 = 18,000\) (move decimal 4 places right)

Rule: Move decimal point to the LEFT

Step 1: Look at the exponent number (ignore the negative sign)

Step 2: Move the decimal point that many places to the left

Step 3: Add zeros if needed

\(4.2 \times 10^{-3} = 0.0042\) (move decimal 3 places left)

\(7.5 \times 10^{-5} = 0.000075\) (move decimal 5 places left)

\(9 \times 10^{-4} = 0.0009\) (move decimal 4 places left)

Calculators display scientific notation in different ways:

E notation: \(3.5E8\) means \(3.5 \times 10^8\)

E notation: \(4.2E-5\) means \(4.2 \times 10^{-5}\)

The "E" stands for "exponent" or "times 10 to the power of"

Method 1: Use the EXP or EE button

To enter \(5.6 \times 10^4\): Type 5.6 → Press EXP → Type 4

Method 2: Use the × 10ˣ button (on some calculators)

Type the coefficient → Press × 10ˣ → Enter the exponent

Display shows: 2.5E12 → Write as: \(2.5 \times 10^{12}\)

Display shows: 3.7E-08 → Write as: \(3.7 \times 10^{-8}\)

Display shows: 6.02E23 → Write as: \(6.02 \times 10^{23}\)

Step 1: Compare the exponents FIRST

• Larger (more positive) exponent → Larger number

• Smaller (more negative) exponent → Smaller number

Step 2: If exponents are equal, compare coefficients

• Larger coefficient → Larger number

✓ Positive exponents are always larger than negative exponents

✓ For positive exponents: \(10^8 > 10^5\)

✓ For negative exponents: \(10^{-2} > 10^{-5}\) (closer to zero is larger)

✓ If exponents equal, compare coefficients normally

Compare: \(5.2 \times 10^7\) and \(3.8 \times 10^9\)
Since \(9 > 7\), therefore \(3.8 \times 10^9 > 5.2 \times 10^7\)

Compare: \(6.5 \times 10^4\) and \(8.2 \times 10^4\)
Exponents equal, compare coefficients: \(8.2 > 6.5\)
Therefore \(8.2 \times 10^4 > 6.5 \times 10^4\)

Compare: \(4.5 \times 10^{-3}\) and \(7.2 \times 10^{-5}\)
Since \(-3 > -5\), therefore \(4.5 \times 10^{-3} > 7.2 \times 10^{-5}\)

The exponents MUST be the same before adding or subtracting!

Step 1: Make sure the exponents are the same

If not, adjust one number by moving the decimal point

Step 2: Add or subtract the coefficients

Step 3: Keep the same exponent

Step 4: Convert back to proper scientific notation if needed

\((a \times 10^n) + (b \times 10^n) = (a + b) \times 10^n\)

\((a \times 10^n) - (b \times 10^n) = (a - b) \times 10^n\)

Example 1: \((3.5 \times 10^4) + (2.8 \times 10^4)\)
Exponents same: \((3.5 + 2.8) \times 10^4 = 6.3 \times 10^4\)

Example 2: \((5.2 \times 10^6) - (1.8 \times 10^6)\)
Exponents same: \((5.2 - 1.8) \times 10^6 = 3.4 \times 10^6\)

Example 3: \((4.5 \times 10^5) + (3.2 \times 10^4)\)
Step 1: Adjust → \((4.5 \times 10^5) + (0.32 \times 10^5)\)
Step 2: Add → \((4.5 + 0.32) \times 10^5 = 4.82 \times 10^5\)

Formula: \((a \times 10^m) \times (b \times 10^n) = (a \times b) \times 10^{m+n}\)

Step 1: Multiply the coefficients (the numbers)

Step 2: Add the exponents

Step 3: Write the result in scientific notation

Step 4: Adjust if the coefficient is not between 1 and 10

Example 1: \((2 \times 10^3) \times (4 \times 10^5)\)
Multiply coefficients: \(2 \times 4 = 8\)
Add exponents: \(3 + 5 = 8\)
Answer: \(8 \times 10^8\)

Example 2: \((3.5 \times 10^6) \times (2 \times 10^{-3})\)
Multiply coefficients: \(3.5 \times 2 = 7\)
Add exponents: \(6 + (-3) = 3\)
Answer: \(7 \times 10^3\)

Example 3: \((5 \times 10^4) \times (6 \times 10^2)\)
Multiply coefficients: \(5 \times 6 = 30\)
Add exponents: \(4 + 2 = 6\)
Result: \(30 \times 10^6\)
Adjust: \(3.0 \times 10^7\) (move decimal left, add 1 to exponent)

Formula: \(\frac{a \times 10^m}{b \times 10^n} = \frac{a}{b} \times 10^{m-n}\)

Step 1: Divide the coefficients (the numbers)

Step 2: Subtract the exponents (top minus bottom)

Step 3: Write the result in scientific notation

Step 4: Adjust if the coefficient is not between 1 and 10

Example 1: \(\frac{8 \times 10^7}{2 \times 10^3}\)
Divide coefficients: \(8 \div 2 = 4\)
Subtract exponents: \(7 - 3 = 4\)
Answer: \(4 \times 10^4\)

Example 2: \(\frac{6 \times 10^5}{3 \times 10^{-2}}\)
Divide coefficients: \(6 \div 3 = 2\)
Subtract exponents: \(5 - (-2) = 5 + 2 = 7\)
Answer: \(2 \times 10^7\)

Example 3: \(\frac{9 \times 10^4}{4 \times 10^6}\)
Divide coefficients: \(9 \div 4 = 2.25\)
Subtract exponents: \(4 - 6 = -2\)
Answer: \(2.25 \times 10^{-2}\)

❌ Mistake 1: Forgetting coefficient must be between 1 and 10

✓ Correct: \(25 \times 10^3\) should be \(2.5 \times 10^4\)

❌ Mistake 2: Adding exponents when adding/subtracting

✓ Correct: Make exponents same first, then add/subtract coefficients

❌ Mistake 3: Subtracting exponents when multiplying

✓ Correct: Multiply → ADD exponents | Divide → SUBTRACT exponents

❌ Mistake 4: Moving decimal wrong direction

✓ Correct: Large number (>10) → positive exponent | Small number (<1) → negative exponent

❌ Mistake 5: Comparing coefficients before exponents

✓ Correct: Always compare exponents first, then coefficients if needed

⚡ Remember: Coefficient between 1-10 | Add/Subtract → Same exponents | Multiply → ADD | Divide → SUBTRACT! ⚡

Astronomy: Distance to stars: \(9.46 \times 10^{15}\) meters (1 light-year)

Biology: Size of a virus: \(1 \times 10^{-7}\) meters

Chemistry: Avogadro's number: \(6.02 \times 10^{23}\) particles/mole

Physics: Speed of light: \(3 \times 10^8\) meters/second

Technology: Computer processing: \(2.5 \times 10^9\) Hz (2.5 GHz)

✓ Master conversions first: Practice converting between standard and scientific notation

✓ Remember exponent rules: Positive for large, negative for small numbers

✓ Check your coefficient: Always between 1 and 10 in final answer

✓ Use your calculator: Learn the EXP or EE button

✓ Practice operations: Start simple, then try complex problems