Operations with
Scientific Notation
Master adding, subtracting, multiplying, and dividing numbers in scientific notation: the universal shorthand for the very large and very small.
8th Grade Math
What Is Scientific Notation?
Scientific notation is a way to write very large or very small numbers compactly. Every number is expressed as a coefficient times a power of ten:
Coefficient \( a \)
A number between \(1\) and \(10\) (not including \(10\))
Exponent \( n \)
Tells how many places the decimal moves: positive = large, negative = small
Why Does It Matter?
Scientists and engineers deal with numbers that span dozens of digits. Scientific notation keeps these manageable and makes computation far simpler.
Standard Form ↔ Scientific Notation
To Scientific Notation
Move the decimal until you have a number between \(1\) and \(10\). Count the places moved: that is your exponent \(n\).
To Standard Form
Move the decimal \(n\) places. Positive \(n\): move right. Negative \(n\): move left. Fill empty spots with zeros.
Multiplying in Scientific Notation
1
Multiply the coefficients: \(a \times b\)
2
Add the exponents: \(m + n\)
3
Adjust if the new coefficient is \(\geq 10\) or \(< 1\)
Dividing in Scientific Notation
1
Divide the coefficients: \( \frac{a}{b} \)
2
Subtract the exponents: \( m - n \)
3
Adjust the coefficient back to proper form if needed
Adding & Subtracting: The Key Rule
1
Check exponents. If they differ, rewrite one term so both share the same power of 10.
2
Add or subtract the coefficients: \( (a \pm b) \times 10^n \)
3
Adjust the result to proper scientific notation if needed.
Key Distinction: Two Families of Operations
| Multiply / Divide | Add / Subtract | |
|---|---|---|
| Coefficients | Multiply or divide them | Add or subtract them |
| Exponents | Add or subtract | Must be equal first |
| Prerequisite | None: works directly | Match exponents first |
| Final step | Adjust coefficient | Adjust coefficient |
How to Apply the Operations
1
Identify the operation: are you multiplying, dividing, adding, or subtracting?
2
For + or −: Make the exponents match by adjusting one coefficient.
3
Operate on coefficients (multiply, divide, add, or subtract).
4
Handle exponents: add (for ×), subtract (for ÷), or keep (for +/−).
5
Adjust to proper form: Ensure \(1 \leq |a| < 10\). Shift decimal and update exponent.
Example 1: Multiplication
1
Multiply coefficients: \( 3.0 \times 2.5 = \) \(7.5\)
2
Add exponents: \( 4 + 3 = \) \(7\)
3
Combine: \( 7.5 \times 10^7 \) (already in proper form since \(1 \leq 7.5 < 10\))
Example 2: Division
1
Divide coefficients: \( 8.4 \div 2.1 = \) \(4.0\)
2
Subtract exponents: \( 9 - 3 = \) \(6\)
3
Combine: \( 4.0 \times 10^6 \)
Example 3: Addition
1
Match exponents. Rewrite \(4.7 \times 10^4\) as \(0.47 \times 10^5\)
2
Add coefficients: \( 3.2 + 0.47 = \) \(3.67\)
3
Combine: \( 3.67 \times 10^5 \) (already proper form)
Example 4: Adjusting the Coefficient
1
Multiply coefficients: \( 6.0 \times 4.0 = \) \(24.0\) (too big!)
2
Add exponents: \( 5 + 3 = 8 \) → gives \(24.0 \times 10^8\)
3
Adjust: \(24.0 = 2.4 \times 10^1\), so \(2.4 \times 10^1 \times 10^8 = \) \(2.4 \times 10^9\)
Reading Calculator Notation
Calculators and technology often display scientific notation differently. You need to recognize and interpret these formats:
3.5E8
Means \(3.5 \times 10^8\). The "E" stands for "exponent."
2.1E−5
Means \(2.1 \times 10^{-5}\). Negative after E means a small number.
7.03 EE 12
Some calculators use "EE" button. This means \(7.03 \times 10^{12}\).
Choosing Appropriate Units
When working with real-world measurements, pick units that keep your numbers in a convenient range. Scientific notation helps you decide.
Seafloor Spreading
About \(2.5 \times 10^{-2}\) m/year, better expressed as 25 mm/year
Galaxy Distance
\(2.5 \times 10^{22}\) m is awkward: use \(2.5\) million light-years
Common Mistakes to Avoid
✕ Adding Exponents When Adding
You cannot add exponents when adding/subtracting. You must match them first, then add the coefficients only.
✕ Forgetting to Adjust
After operating, if the coefficient is \(\geq 10\) or \(< 1\), you must shift the decimal and update the exponent.
✕ Wrong Exponent Sign
Subtracting a negative exponent: \(5 - (-3) = 8\), not \(2\). Watch your signs carefully.
Question 1 of 5
What is \( (4.0 \times 10^3) \times (2.0 \times 10^5) \)?
Multiply the coefficients: \(4.0 \times 2.0 = 8.0\). Add the exponents: \(3 + 5 = 8\). The answer is \(8.0 \times 10^8\).
Question 2 of 5
What is \( \dfrac{9.6 \times 10^7}{3.2 \times 10^4} \)?
Divide the coefficients: \(9.6 \div 3.2 = 3.0\). Subtract the exponents: \(7 - 4 = 3\). The answer is \(3.0 \times 10^3\).
Question 3 of 5
What is \( 5.3 \times 10^6 + 2.1 \times 10^5 \)?
Rewrite \(2.1 \times 10^5\) as \(0.21 \times 10^6\). Then add: \(5.3 + 0.21 = 5.51\). The answer is \(5.51 \times 10^6\).
Question 4 of 5
A calculator displays 6.02E23. Which of the following is the correct interpretation?
The "E" on a calculator means "times ten to the power of." So 6.02E23 = \(6.02 \times 10^{23}\). Choice C has an improper coefficient (\(602\) is not between 1 and 10).
Question 5 of 5
What is \( (7.2 \times 10^4) \times (5.0 \times 10^{-2}) \) expressed in proper scientific notation?
Multiply coefficients: \(7.2 \times 5.0 = 36.0\). Add exponents: \(4 + (-2) = 2\). That gives \(36.0 \times 10^2\), which is not proper form. Adjust: \(36.0 = 3.6 \times 10^1\), so \(3.6 \times 10^1 \times 10^2 = \) \(3.6 \times 10^3\).