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\(10\)
\(n\)

Operations with
Scientific Notation

Learn to multiply, divide, add, and subtract numbers expressed in scientific notation and interpret the results with proper units.

Pre-Algebra

What Is Scientific Notation?

Scientific notation is a way to write very large or very small numbers compactly. Instead of writing all the zeros, we use a coefficient multiplied by a power of 10.

Large Number
\( 93{,}000{,}000 = 9.3 \times 10^7 \)
Small Number
\( 0.00042 = 4.2 \times 10^{-4} \)

Why Does It Matter?

Scientists, engineers, and mathematicians constantly work with numbers that are extremely large or tiny. Scientific notation makes these numbers easier to read, compare, and compute with.

\(10^{-8}\)
Virus
\(10^{-3}\)
grain of sand
\(10^{0}\)
human
\(10^{7}\)
Earth
\(10^{26}\)
universe
Scale of Powers of 10 (meters)

Anatomy of \( a \times 10^n \)

\( \color{#4ade80}6.02 \color{#e2e8f0}\times \color{#818cf8}10^{\color{#f87171}23} \)
coefficient
\( 1 \le |a| < 10 \)
base
always 10
exponent
integer (+ or −)

Multiplying in Scientific Notation

1
Multiply the coefficients: \( a \times b \)
2
Add the exponents: \( m + n \)
3
Adjust if the new coefficient is not between 1 and 10.

Dividing in Scientific Notation

1
Divide the coefficients: \( \dfrac{a}{b} \)
2
Subtract the exponents: \( m - n \)
3
Adjust the coefficient if needed so it falls in \( [1, 10) \).

Multiply vs. Divide

Multiplication Division
Coefficients Multiply: \( a \times b \) Divide: \( a \div b \)
Exponents Add: \( m + n \) Subtract: \( m - n \)
Final Step Adjust coefficient to \( [1, 10) \) if needed
Example \( 3 \times 2 = 6 \), exponent \( 4+3=7 \) \( 6 \div 2 = 3 \), exponent \( 7-3=4 \)

Adding & Subtracting

1
Make exponents the same: Adjust the smaller-exponent term by shifting its decimal.
2
Add or subtract the coefficients. Keep the shared \( 10^n \).
3
Adjust the result back to proper scientific notation.

Step-by-Step Approach

1
Identify the operation: Are you multiplying, dividing, adding, or subtracting?
2
For × or ÷: Handle coefficients and exponents separately (multiply/add or divide/subtract).
3
For + or −: Match the exponents first, then combine the coefficients.
4
Adjust: Make sure the coefficient is in \( [1, 10) \). Shift the decimal and change the exponent.
5
Interpret units: Attach the correct unit (meters, seconds, grams, etc.) to your final answer.

Example 1: Multiplication

Problem: A bacterium is \( 3.5 \times 10^{-6} \) meters long. A colony has \( 2.0 \times 10^{8} \) bacteria end to end. What is the total length?

\( (3.5 \times 10^{-6}) \times (2.0 \times 10^{8}) \)
Coefficients: \( 3.5 \times 2.0 = 7.0 \)
Exponents: \( -6 + 8 = 2 \)
\( = 7.0 \times 10^2 = 700 \) meters

Example 2: Division

Problem: Light travels \( 9.46 \times 10^{15} \) meters in one year. If one AU is \( 1.496 \times 10^{11} \) meters, how many AU are in a light-year?

\( (9.46 \times 10^{15}) \div (1.496 \times 10^{11}) \)
Coefficients: \( 9.46 \div 1.496 \approx 6.32 \)
Exponents: \( 15 - 11 = 4 \)
\( \approx 6.32 \times 10^4 \) AU

Example 3: Addition

Problem: A town uses \( 4.2 \times 10^6 \) gallons of water in June and \( 3.8 \times 10^5 \) gallons in July. Find the total.

1
Match exponents (use the larger one): rewrite \( 3.8 \times 10^5 \) as \( 0.38 \times 10^6 \).
2
Add coefficients: \( 4.2 + 0.38 = 4.58 \).
3
Result: \( 4.58 \times 10^6 \) gallons. Already in proper form!

Example 4: Adjustment Step

Problem: \( (8.4 \times 10^5) \times (6.0 \times 10^3) \)

Step 1: Coefficients: \( 8.4 \times 6.0 = 50.4 \)
Step 2: Exponents: \( 5 + 3 = 8 \)
\( 50.4 \times 10^8 \) → NOT proper form! (\( 50.4 \ge 10 \))
\( = 5.04 \times 10^9 \)

Interpreting Units

When you compute with scientific notation in context, units follow the same rules as regular arithmetic. Pay attention to what the units become after the operation.

Multiplication
Units multiply too: \( \text{meters} \times \text{meters} = \text{meters}^2 \), \( \text{seconds} \times \frac{\text{meters}}{\text{second}} = \text{meters} \)
Division
Units divide: \( \frac{\text{meters}}{\text{seconds}} \), \( \frac{\text{dollars}}{\text{item}} \)
Addition/Subtraction
Units must be the same: you can only add \(\text{meters}\) to \(\text{meters}\), \(\text{grams}\) to \(\text{grams}\)

Common Mistakes to Avoid

✕ Adding Exponents When Adding
\( 2 \times 10^3 + 3 \times 10^3 \neq 5 \times 10^6 \). You add coefficients only, keeping the shared exponent.
✕ Forgetting to Adjust
After multiplying, if you get \( 36 \times 10^4 \), you must convert: \( 3.6 \times 10^5 \). The coefficient must be in \( [1, 10) \).
✕ Mismatched Exponents
Adding \( 5 \times 10^6 + 3 \times 10^4 \) as \( 8 \times 10^6 \) is wrong. First convert: \( 5 \times 10^6 + 0.03 \times 10^6 = 5.03 \times 10^6 \).

Summary of All Operations

Operation Coefficients Exponents Extra Step
Multiply Multiply Add Adjust if needed
Divide Divide Subtract Adjust if needed
Add Add Must match first Adjust if needed
Subtract Subtract Must match first Adjust if needed

Question 1 of 5

What is \( (3.0 \times 10^4) \times (2.0 \times 10^5) \)?

Question 2 of 5

Simplify: \( \dfrac{8.4 \times 10^{9}}{2.1 \times 10^{3}} \)

Question 3 of 5

What is \( 5.2 \times 10^7 + 3.1 \times 10^6 \)?

Question 4 of 5

A factory produces \( 7.5 \times 10^3 \) widgets per hour. If it runs for \( 4.0 \times 10^2 \) hours, how many widgets are produced in total?

Question 5 of 5

The mass of the Earth is about \( 5.97 \times 10^{24} \) kg, and the mass of the Moon is about \( 7.35 \times 10^{22} \) kg. Approximately how many times more massive is the Earth than the Moon?