Multiplying Scientific Notation

Welcome back to Scientific Notation in Real Life! This is Lesson 5 of 6, which means we are nearly at the finish line. Over the previous four lessons, we built a strong set of skills: reading scientific notation, writing both large and small numbers, and comparing values by examining exponents and coefficients. All of that work focused on understanding and organizing numbers.

Now we move to computing with them. In this lesson, we will learn how to multiply two numbers in scientific notation. This is one of the most practical operations in science and everyday estimation, and the method is refreshingly simple: multiply the coefficients, add the exponents, and adjust if needed. Let's get started.

Many real-world questions boil down to multiplying two large (or small) quantities. For instance, if a city of 350,000 people each uses about 200 liters of water per day, how much water does the entire city need? We could multiply $350{,}000 \times 200$ and count zeros carefully, but that is slow and error-prone.

In scientific notation, those numbers become $3.5 \times 10^5$ and $2.0 \times 10^2$. As we will see shortly, multiplying them takes just a few small steps with no long strings of zeros in sight. Scientific notation keeps the arithmetic compact and the results easy to read.

You may recall from the earlier course on exponent rules that multiplying two powers of the same base means we add the exponents:

Every number in scientific notation has the form $c \times 10^n$. When we multiply two such numbers, we can group the coefficients together and the powers of ten together:

So the procedure is: multiply the coefficients and add the exponents. That is the heart of the method.

Let's walk through a multiplication where everything lands neatly in proper form:

• Multiply the coefficients: $3 \times 2 = 6$
• Add the exponents: $4 + 3 = 7$
• Combine: $6 \times 10^7$

Since $6$ is between 1 and 10, the result is already in proper scientific notation. Done!

Here is one more:

Coefficients: $4 \times 2 = 8$. Exponents: $2 + 5 = 7$. Result: $8 \times 10^7$. The coefficient $8$ sits comfortably between 1 and 10, so no extra work is needed.

Sometimes the product of the two coefficients lands at 10 or above. Recall that proper scientific notation requires the coefficient to be at least 1 and less than 10. When the product breaks that rule, we need one extra step.

Consider:

• Multiply the coefficients: $5 \times 4 = 20$
• Add the exponents: $3 + 2 = 5$
• Combine: $20 \times 10^5$

The coefficient $20$ is too large, so this is not proper form yet. To fix it, we move the decimal point one place to the left, turning $20$ into $2.0$, and add 1 to the exponent to compensate:

Why does this work? Because $20 = 2.0 \times 10^1$, and $10^1 \times 10^5 = 10^6$. We are simply rewriting the same value in proper form.

Let's try another example with decimal coefficients:

Coefficients: $3.5 \times 6.0 = 21.0$. Exponents: $4 + 2 = 6$. Intermediate result: $21.0 \times 10^6$. Since $21.0 \geq 10$, we shift the decimal one place left and increase the exponent by 1:

Here is the complete method gathered into four tidy steps:

1. Multiply the coefficients ($c_1 \times c_2$).
2. Add the exponents ($n_1 + n_2$).
3. Combine the results: $(\text{product}) \times 10^{(\text{sum})}$.
4. If the product is 10 or greater, move the decimal one place left and add 1 to the exponent.

A quick note on Step 4: since both original coefficients fall between 1 and 10, their product can never reach 100. This means we will never need to shift the decimal more than one place. The adjustment step is always a small fix, not a large overhaul.

The table below shows three multiplications side by side so we can compare the "no adjustment" and "adjustment" cases:

Let's put the method to work on a practical problem. Suppose a city has a population of $3.5 \times 10^5$ people, and each person uses an average of $2.0 \times 10^2$ liters of water per day. What is the city's total daily water consumption?

• Coefficients: $3.5 \times 2.0 = 7.0$
• Exponents: $5 + 2 = 7$
• Result: $7.0 \times 10^7$ liters

Since $7.0$ is between 1 and 10, no adjustment is needed. The city uses about 70 million liters of water per day. Notice how scientific notation kept the calculation to small, manageable numbers while still handling quantities in the tens of millions.

A few common mistakes are worth keeping in mind as you start practicing:

• Adding the exponents is correct; multiplying them is not. The product rule says $10^a \times 10^b = 10^{a+b}$. It is easy to slip into $a \times b$ by habit — always add.
• Forgetting to adjust the coefficient. If the product of the coefficients is 10 or more, the answer is not yet in proper scientific notation. The same is true if it falls below 1 — always verify the coefficient lands in the range $[1, 10)$ before finishing.
• Rushing through decimal arithmetic. A small error in a coefficient multiplication like $3.5 \times 6.0$ can throw off the entire answer. Take a moment to double-check before moving on.

In this lesson, we learned how to multiply two numbers in scientific notation: multiply the coefficients, add the exponents, and adjust the result into proper form when the new coefficient reaches 10 or above. We also applied the method to estimate a city's daily water consumption, seeing firsthand how scientific notation keeps large-number arithmetic quick and clean.

Up next, you will practice this skill through a series of hands-on exercises — from guided step-by-step problems to real-world scenarios involving city water supplies and the speed of light. Work through them confidently, and this multiplication technique will soon feel like second nature!