Multiplying Numbers In Scientific Notation

Download as PDF

Ready to tackle huge numbers like a pro? This lesson shows you the super-simple trick for multiplying numbers in scientific notation. You'll learn how to handle the coefficients and the exponents to get the right answer every time, making big math problems feel small!

What Is Scientific Notation?

Multiplying numbers in scientific notation is a process used to find the product of two numbers that are written in a special format. Before we multiply, let's quickly review what scientific notation is. It's a way of writing very large or very small numbers easily using powers of 10. A number in scientific notation looks like this:

a \times 10^{n}

This expression has two main parts:

The coefficient (the $a$ ): This is a number that must be greater than or equal to $1$ but less than $10$ . For example, $3.5$ or $7$ or $9.99$ .
The power of 10 (the $10^{n}$ ): This part tells us how many places to move the decimal point. The $n$ is called the exponent. If the exponent is positive, it represents a large number. If it's negative, it represents a small number (a decimal).

For example, the distance to the Sun is about $93, 000, 000$ miles. In scientific notation, we write this as $9.3 \times 10^{7}$ miles. It's much shorter and easier to work with!

How Do You Multiply Numbers in Scientific Notation?

Multiplying in scientific notation might look complicated, but it's based on a simple, three-step process. Let's say we want to multiply two numbers:

$(a \times 10^{m})$ and $(b \times 10^{n})$

Here are the steps to follow:

Multiply the Coefficients: Take the first numbers from each expression (the $a$ and $b$ ) and multiply them together. So, you'll calculate $a \times b$ .
Add the Exponents: Take the exponents from the powers of 10 (the $m$ and $n$ ) and add them together. This is because when you multiply powers with the same base (in this case, 10), you add their exponents. So, you'll have $10^{m + n}$ .
Combine and Adjust: Put your new coefficient and new power of 10 together. Your result will look like $(a \times b) \times 10^{m + n}$ . The final, most important step is to check if your new coefficient is between $1$ and $10$ . If it's not, you'll need to adjust it, which we'll show you how to do in the examples.

This method works because of the commutative property of multiplication, which says you can reorder numbers when you multiply. We can regroup the problem like this:

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

Let's see it in action with some real numbers!

Worked Example 1: A Straightforward Multiplication

In our first example, the product of the coefficients will already be in the correct range, making it a great place to start.

Example 1

Calculate $(2 \times 10^{5}) \times (4 \times 10^{3})$ .

Step 1: Multiply the coefficients.

The coefficients are $2$ and $4$ .

$2 \times 4 = 8$

Step 2: Add the exponents.

The exponents are $5$ and $3$ .

$5 + 3 = 8$

So, our new power of 10 is $10^{8}$ .

Step 3: Combine and check.

Put the new coefficient and the new power of 10 together.

$8 \times 10^{8}$

Now, we check: is the coefficient $8$ between $1$ and $10$ ? Yes, it is! So, our final answer is in proper scientific notation.

Answer: $(2 \times 10^{5}) \times (4 \times 10^{3}) = 8 \times 10^{8}$

Worked Example 2: When You Need to Adjust the Answer

Sometimes, when you multiply the coefficients, you'll get a number that is $10$ or larger. When this happens, you need to perform an extra step to put the answer back into proper scientific notation.

Example 2

Calculate $(6 \times 10^{4}) \times (5 \times 10^{2})$ .

Step 1: Multiply the coefficients.

The coefficients are $6$ and $5$ .

$6 \times 5 = 30$

Step 2: Add the exponents.

The exponents are $4$ and $2$ .

$4 + 2 = 6$

Our new power of 10 is $10^{6}$ .

Step 3: Combine and adjust.

Putting our results together gives us $30 \times 10^{6}$ . Now, we must check the coefficient. The number $30$ is not between $1$ and $10$ , so this is not proper scientific notation. We need to fix it!

To adjust, we rewrite the coefficient $30$ in scientific notation. We move the decimal point one place to the left to get $3.0$ . Since we made the number smaller, we have to account for that with a power of 10. So, $30 = 3.0 \times 10^{1}$ .

Now, substitute this back into our expression:

$(3.0 \times 10^{1}) \times 10^{6}$

We have two powers of 10, so we add their exponents again!

$3.0 \times 10^{1 + 6} = 3.0 \times 10^{7}$

Our new coefficient $3.0$ is between $1$ and $10$ . Now we are done.

Answer: $(6 \times 10^{4}) \times (5 \times 10^{2}) = 3 \times 10^{7}$

Worked Example 3: Multiplying with Negative Exponents

Scientific notation is also used for very small numbers, which have negative exponents. The multiplication process is exactly the same, but you need to be careful when adding negative numbers.

Example 3

Calculate $(9.1 \times 10^{- 5}) \times (2 \times 10^{- 3})$ .

Step 1: Multiply the coefficients.

The coefficients are $9.1$ and $2$ .

$9.1 \times 2 = 18.2$

Step 2: Add the exponents.

The exponents are $- 5$ and $- 3$ . Be careful with the signs!

$(- 5) + (- 3) = - 8$

Our new power of 10 is $10^{- 8}$ .

Step 3: Combine and adjust.

Our intermediate result is $18.2 \times 10^{- 8}$ . Let's check the coefficient. The number $18.2$ is not between $1$ and $10$ , so we must adjust it.

We rewrite $18.2$ in scientific notation: $18.2 = 1.82 \times 10^{1}$ .

Now, substitute this back into our expression:

$(1.82 \times 10^{1}) \times 10^{- 8}$

Finally, combine the powers of 10 by adding their exponents:

$1.82 \times 10^{1 + (- 8)} = 1.82 \times 10^{- 7}$

The coefficient $1.82$ is now in the correct range.

Answer: $(9.1 \times 10^{- 5}) \times (2 \times 10^{- 3}) = 1.82 \times 10^{- 7}$

What Are Some Common Mistakes to Avoid?

When you're learning this process, it's easy to make small mistakes. Here are the most common ones to watch out for:

Forgetting to Adjust the Coefficient: A very common error is to finish step 2 and forget to check if the answer is in proper scientific notation. An answer like $54 \times 10^{8}$ is mathematically correct but isn't finished yet. Always make sure your final coefficient is between $1$ and $10$ .
Multiplying the Exponents: Remember, the rule for multiplying powers (like $10^{5} \times 10^{3}$ ) is to add the exponents, not multiply them. You should get $10^{8}$ , not $10^{15}$ .
Adding the Coefficients: Students sometimes get the two steps mixed up. You must multiply the coefficients (the front numbers) and add the exponents (the powers of 10).
Sign Errors with Negative Exponents: Be extra careful when adding negative exponents. For example, $10^{7} \times 10^{- 3}$ becomes $10^{7 + (- 3)} = 10^{4}$ . And $10^{- 4} \times 10^{- 2}$ becomes $10^{- 4 + (- 2)} = 10^{- 6}$ . Reviewing rules for adding integers can help!

The table below summarizes the correct operations versus the common mistakes.

Part of the Problem	Correct Operation	Common Mistake
Coefficients (e.g., $3.1$ and $2$ )	Multiply: $3.1 \times 2 = 6.2$	Adding: $3.1 + 2 = 5.1$
Exponents (e.g., $10^{4}$ and $10^{5}$ )	Add: $4 + 5 = 9$ for $10^{9}$	Multiplying: $4 \times 5 = 20$ for $10^{20}$

Quick Summary: The Step-by-Step Rule

Feeling confident? Here is a quick summary of the entire process for multiplying $(a \times 10^{m}) \times (b \times 10^{n})$ . You can use this as a quick reference.

(First Numbers)

\times

(First Numbers) and (Exponents)

+

(Exponents)

Multiply Coefficients: Calculate $a \times b$ .
Add Exponents: Calculate $m + n$ .
Combine: Write down your temporary answer: $(a \times b) \times 10^{m + n}$ .
Adjust (if needed): Check if the new coefficient $(a \times b)$ is between $1$ and $10$ . If not, rewrite it in scientific notation and combine the new power of 10 with the existing one by adding the exponents.

Practice is the key to mastering this skill. The more you do it, the more natural it will become, and soon you'll be multiplying giant and tiny numbers in your head!

Frequently Asked Questions

Why do you add the exponents instead of multiplying them?

This rule comes from a fundamental property of exponents. When you multiply two powers that have the same base, you add their exponents. For example, $10^{2} \times 10^{3}$ is $(10 \times 10) \times (10 \times 10 \times 10)$ , which is five 10s multiplied together, or $10^{5}$ . Notice that $2 + 3 = 5$ .

What do I do if one of the numbers isn't in scientific notation?

Before you can use this method, both numbers must be in proper scientific notation. You should first convert the number that isn't in the correct format. For example, to multiply $4, 500 \times (2 \times 10^{3})$ , you would first change $4, 500$ to $4.5 \times 10^{3}$ and then proceed.

Does this method work for division too?

The process for division is very similar, but the operations are reversed. To divide numbers in scientific notation, you divide the coefficients and subtract the exponents. For example, $(8 \times 10^{5}) \div (2 \times 10^{2}) = (8 \div 2) \times 10^{5 - 2} = 4 \times 10^{3}$ .

What happens if an exponent is zero?

An exponent of zero doesn't change anything! Remember that any number to the power of zero is $1$ , so $10^{0} = 1$ . When you add the exponents, adding zero won't change the other exponent. For instance, $(3 \times 10^{5}) \times (2 \times 10^{0}) = 6 \times 10^{5 + 0} = 6 \times 10^{5}$ .

Can my final coefficient be exactly 1?

Yes, it can. This often happens when the product of the original coefficients is a power of 10, like 10 or 100. For example, $(5 \times 10^{3}) \times (2 \times 10^{4})$ gives you $10 \times 10^{7}$ . After adjusting, this becomes $(1 \times 10^{1}) \times 10^{7} = 1 \times 10^{8}$ .

Is it okay to use a calculator for this?

While a calculator can give you the final answer, it's very important to learn these steps by hand. Understanding the process helps you grasp how exponents work and allows you to check if your calculator's answer makes sense. Many tests will require you to show your work.

How does adjusting the coefficient work again?

When you adjust, you are just splitting a number into two parts. If you have $25 \times 10^{4}$ , you can rewrite $25$ as $2.5 \times 10^{1}$ . You made the coefficient smaller by a factor of 10, so you must increase the power of 10 by one to keep the overall value the same. This gives you $(2.5 \times 10^{1}) \times 10^{4} = 2.5 \times 10^{5}$ .