Write 85,000 Using Factors Of 10

Understanding place value and factors of ten is a fundamental concept in mathematics. When we talk about expressing a number like 85,000 as a product with 10 as a factor, we're essentially breaking it down into its multiplicative components. This skill is super useful for understanding large numbers, simplifying calculations, and grasping concepts like scientific notation. Let's dive into how we can represent 85,000 in this specific way and explore why one of the given options is the correct mathematical expression. We'll be looking at the options:

• A. $85 \times 10 \times 10$
• B. $85 \times 10 \times 10 \times 10$
• C. $85 \times 10 \times 10 \times 10 \times 10$

To figure this out, we need to think about what multiplying by 10 actually does to a number. Each time you multiply a number by 10, you essentially add a zero to the end of it, or shift all the digits one place to the left and add a zero in the ones place. So, if we start with 85, multiplying by 10 gives us 850. Multiplying by 10 again (which is the same as multiplying by $10 \times 10$, or 100) gives us 8,500. If we multiply by 10 a third time (multiplying by $10 \times 10 \times 10$, or 1,000), we get 85,000.

Let's test this:

• Option A: $85 \times 10 \times 10$ This is the same as $85 \times 100$. If we calculate this, we get 8,500. This is not equal to 85,000, so option A is incorrect.

• Option B: $85 \times 10 \times 10 \times 10$ This is the same as $85 \times 1,000$. If we calculate this, we get 85,000. This matches our target number, so option B is a strong contender.

• Option C: $85 \times 10 \times 10 \times 10 \times 10$ This is the same as $85 \times 10,000$. If we calculate this, we get 850,000. This is much larger than 85,000, so option C is incorrect.

Therefore, the expression that correctly shows how you can write the number 85,000 as a product with 10 as a factor is Option B: $85 \times 10 \times 10 \times 10$. This breaks down 85,000 into 85 multiplied by three factors of 10, which is equivalent to $85 \times 10^3$ or $85 \times 1,000$. It's a fantastic way to visualize the magnitude of the number and reinforces the power of ten in our number system. Keep practicing these kinds of problems, and you'll become a math whiz in no time!

The Power of Place Value

The concept of place value is absolutely central to understanding how numbers are constructed and manipulated in mathematics. When we write a number like 85,000, each digit holds a specific value based on its position. The '8' is in the ten thousands place, meaning it represents $8 \times 10,000$. The '5' is in the thousands place, representing $5 \times 1,000$. The zeros that follow indicate that there are no hundreds, tens, or ones.

Expressing 85,000 as $85 \times 10 \times 10 \times 10$ is a direct application of this place value understanding. We can think of 85,000 as 85 thousands. The word 'thousands' itself implies multiplication by 1,000. And as we've seen, 1,000 is precisely $10 \times 10 \times 10$. So, 85 thousands is the same as 85 times 1,000, which can be written as $85 \times (10 \times 10 \times 10)$.

This method is particularly helpful when dealing with larger numbers or when transitioning to scientific notation. Scientific notation is a way of expressing numbers that are too large or too small to be conveniently written in decimal form. It is typically written in the form $a \times 10^n$, where $1 le a < 10$ and $n$ is an integer. In our case, 85,000 can be written in scientific notation as $8.5 \times 10^4$. To see how this relates to our problem, let's expand $8.5 \times 10^4$:

$8.5 \times 10^4 = 8.5 \times (10 \times 10 \times 10 \times 10) = 8.5 \times 10,000 = 85,000$.

Now, let's compare this to our chosen answer, $85 \times 10 \times 10 \times 10$:

$85 \times 10 \times 10 \times 10 = 85 \times 1,000 = 85,000$.

While both expressions correctly equal 85,000, the question specifically asks to write it as a product with 10 as a factor. This implies we should be looking for expressions where 10 is explicitly shown as a repeated multiplier, as in $85 \times 10^n$. Option B, $85 \times 10 \times 10 \times 10$, clearly demonstrates this by showing three explicit factors of 10.

Understanding this connection between place value, factors of ten, and scientific notation is a cornerstone of numerical literacy. It allows us to comprehend the vastness of numbers encountered in science, economics, and everyday life. By breaking down large numbers into manageable components, we can perform calculations more efficiently and gain a deeper appreciation for the structure of our number system. The more you practice these exercises, the more intuitive these relationships will become, making complex mathematical concepts feel much simpler.

Why Other Options Don't Measure Up

Let's take a moment to thoroughly examine why the other options, A and C, are not the correct way to express 85,000 as a product with 10 as a factor. This detailed analysis will reinforce our understanding of multiplication and place value, ensuring we're completely confident in our final answer.

Option A: $85 \times 10 \times 10$

This expression simplifies to $85 \times 100$. When we perform this multiplication, we are essentially taking the number 85 and multiplying it by one hundred. Visually, this means adding two zeros to the end of 85, resulting in 8,500. Clearly, 8,500 is not equal to 85,000. This option only includes two factors of 10. To reach 85,000, we need to account for the thousands place, which requires multiplying by 1,000 (three factors of 10). Therefore, option A falls short by a factor of 10.

Option C: $85 \times 10 \times 10 \times 10 \times 10$

This expression simplifies to $85 \times 10,000$. Here, we are multiplying 85 by ten thousand. Performing this calculation results in 850,000. This number is significantly larger than our target number of 85,000. In fact, it's ten times larger. This option includes four factors of 10. This indicates that we have included one too many factors of 10 in this expression. To get back to 85,000 from 850,000, we would need to divide by 10, or in terms of factors, remove one factor of 10.

The Importance of Precision in Mathematical Expressions

These examples highlight the critical importance of precision in mathematics. Each factor of 10 contributes a significant magnitude to the final number. When we are asked to represent a specific value, like 85,000, it's not enough for an expression to be close; it must be exact. The number of times we multiply by 10 directly corresponds to the number of zeros that follow the non-zero digits in the number (when the leading number is between 1 and 9.99...).

In the case of 85,000, the '8' is in the ten thousands place, and the '5' is in the thousands place. This structure tells us we are dealing with a number that is 85 units of a thousand. Since a thousand is $10 \times 10 \times 10$, the number 85,000 can be accurately represented as $85 \times (10 \times 10 \times 10)$.

By understanding why options A and C are incorrect, we gain a deeper appreciation for how multiplication by powers of 10 expands numbers. It solidifies the understanding that each additional factor of 10 effectively increases the number's value by a factor of ten. This meticulous approach to problem-solving ensures that we not only arrive at the correct answer but also build a robust foundation for future mathematical endeavors. Always double-check the number of factors and their contribution to the overall value!

Conclusion: The Correct Expression Revealed

After carefully analyzing the structure of the number 85,000 and the effect of multiplying by factors of 10, we can definitively conclude which expression is correct. The question asks us to represent 85,000 as a product where 10 is a factor. This means we need to find an expression of the form $85 \times 10 \times 10 \times ... \times 10$ that equals exactly 85,000.

We tested each option:

• Option A, $85 \times 10 \times 10$, equals $85 \times 100 = 8,500$. This is too small.
• Option B, $85 \times 10 \times 10 \times 10$, equals $85 \times 1,000 = 85,000$. This is the exact value.
• Option C, $85 \times 10 \times 10 \times 10 \times 10$, equals $85 \times 10,000 = 850,000$. This is too large.

Therefore, the correct expression is B. $85 \times 10 \times 10 \times 10$. This expression breaks down 85,000 into its core components: the number 85 and three factors of 10, which together form the value of one thousand. This demonstrates a clear understanding of how multiplication and place value work together to create our number system.

Mastering these fundamental concepts will serve you well as you continue your mathematical journey. Understanding how numbers are built and how operations like multiplication affect them is key to tackling more complex problems. Keep practicing, and don't hesitate to explore further!

For more on place value and number operations, you can visit the National Council of Teachers of Mathematics (NCTM) website, a fantastic resource for educators and students alike.