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Learn to add, subtract, multiply, and divide very large and very small numbers using the powerful shorthand of scientific notation.
Imagine trying to write out the distance from Earth to the nearest star — about 40,000,000,000,000 kilometers. That's a lot of zeros! Now imagine a scientist measuring a virus that's 0.000000120 meters wide. Writing all those zeros over and over is tiring and makes it easy to make mistakes. Scientific notation was invented to solve exactly this problem — it's a compact way to write very big or very small numbers.
Throughout history, mathematicians and scientists kept running into numbers that were just too large or too small to handle easily. Here's how the story unfolded:
The big question this lesson tackles is: once you've written numbers in scientific notation, how do you actually do math with them? Can you add two numbers in scientific notation? Multiply them? What if one number is in scientific notation and the other is a regular decimal? Let's find out.
Before we start doing operations, let's make sure we're solid on what scientific notation actually is and the key ideas that make it work.
One of the coolest things about scientific notation is how it lets you see the scale of numbers. Each power of ten is ten times bigger than the one before it. The diagram below shows where different powers of ten sit on a number line, with real-world examples to give you a feel for the sizes.
Notice how the powers of ten spread across a huge range. Between 10⁻⁸ (the size of a virus) and 10¹¹ (the distance to the Sun), there are 19 powers of ten! Scientific notation lets you work with all these scales in one compact format. When you do math with these numbers, the exponent tells you which "zoom level" you're at.
Now for the main event! There are four operations to master: multiplication, division, addition, and subtraction. Each one has its own set of rules. Let's go through them one at a time.
Here's why this works: when you multiply powers of ten, you add the exponents. For example, 10³ × 10⁴ = 10⁷ (because 1,000 × 10,000 = 10,000,000). So you just multiply the front numbers (the coefficients) and add the exponents together. If your answer's coefficient isn't between 1 and 10, adjust it.
Division is the opposite of multiplication. When you divide powers of ten, you subtract the exponents. So 10⁸ ÷ 10³ = 10⁵. Divide the coefficients normally, subtract the exponents, and adjust if needed.
This is the trickiest one! You cannot add or subtract two numbers in scientific notation unless they have the same power of ten. If the exponents are different, you need to adjust one of the numbers first. It's just like adding fractions — you need a common denominator. Here, you need a common exponent.
Sometimes a problem gives you one number in scientific notation and another as a regular decimal. Your first step is always to convert them to the same format. Usually, it's easiest to convert the decimal into scientific notation. For example, if you're asked to multiply 4,500 × (2 × 10³), first rewrite 4,500 as 4.5 × 10³, then multiply normally.
Different operations follow different paths. This flowchart shows you the decision process: first figure out what operation you need, then follow the correct steps.
The Final Check at the bottom is super important. After any operation, your answer might have a coefficient like 32.5 or 0.48. Those aren't in proper scientific notation! You need to adjust: if the coefficient is 32.5, rewrite it as 3.25 and increase the exponent by 1. If it's 0.48, rewrite it as 4.8 and decrease the exponent by 1.
| Operation | What to Do with Coefficients | What to Do with Exponents | Example |
|---|---|---|---|
| Multiply | Multiply them | Add them | (3 × 10⁴)(2 × 10³) = 6 × 10⁷ |
| Divide | Divide them | Subtract them | (8 × 10⁶) ÷ (4 × 10²) = 2 × 10⁴ |
| Add | Add them (match exponents first) | Keep them the same | 3 × 10⁵ + 2 × 10⁵ = 5 × 10⁵ |
| Subtract | Subtract them (match exponents first) | Keep them the same | 7 × 10⁴ − 3 × 10⁴ = 4 × 10⁴ |
Let's work through a complete problem that combines both decimal and scientific notation, step by step. This is the kind of problem you'll see on tests and homework.
(3.6 × 10⁴) × (2.4 × 10⁻⁵)You might wonder: why not just use regular decimals all the time? Or why not always use scientific notation? Each format has strengths and weaknesses. Here's a comparison to help you decide.
| Feature | Decimal (Standard) Form | Scientific Notation |
|---|---|---|
| Readability | Easy to read for everyday numbers (like 42 or 3.5) | Easier for very large or very small numbers |
| Multiplying | Can be tedious with many zeros | Quick — just multiply coefficients and add exponents |
| Adding | Line up the decimal points — straightforward | Must match exponents first — extra step required |
| Error risk | Easy to lose count of zeros in big numbers | Compact format reduces zero-counting errors |
| Best used for | Numbers you encounter in daily life | Science, engineering, astronomy, microbiology |
The bottom line? Multiplication and division are actually easier in scientific notation. You just work with the coefficients and exponents separately. But addition and subtraction require that extra step of matching exponents, which can feel a bit tricky at first. With practice, though, it becomes second nature.
What you're learning right now is the foundation for many advanced topics. Scientific notation isn't just a school exercise — it's a tool that real scientists and engineers use every day. Here's how it connects to what comes next.
| What You Learn Now | Where It Leads |
|---|---|
| Adding exponents when multiplying | Exponent rules in Algebra 1 and beyond (like x³ × x⁴ = x⁷) |
| Subtracting exponents when dividing | Simplifying expressions with variables and exponents |
| Adjusting coefficients to proper form | Significant figures in high school chemistry and physics |
| Mixing decimal and scientific notation | Unit conversions in science — moving between metric prefixes (kilo, milli, micro, nano) |
| Working with negative exponents | Logarithms — a way to work with exponents in reverse |
In high school science classes, you'll constantly see numbers like 6.022 × 10²³ (Avogadro's number in chemistry) or 3.0 × 10⁸ m/s (the speed of light in physics). Being comfortable doing operations with these numbers will make those classes so much smoother. You're building a skill right now that you'll use for years.
Time to test your skills! Try each problem on your own before clicking "Show Answer." The problems get harder as you go.
Scientific notation writes numbers as a coefficient (between 1 and 10) multiplied by a power of ten. When you multiply two numbers in scientific notation, you multiply the coefficients and add the exponents. When you divide, you divide the coefficients and subtract the exponents. For addition and subtraction, you must first make the exponents match — just like finding a common denominator — then add or subtract only the coefficients while keeping the exponent the same.
When a problem gives you numbers in mixed formats (some in decimal form, some in scientific notation), your first step is always to convert everything into the same format. After any operation, always do a final check: is your coefficient between 1 and 10? If not, adjust by moving the decimal point and changing the exponent accordingly. With these rules in your toolkit, you can confidently handle numbers of any size — from atoms to galaxies.