Equivalent Expression To 3 X 5 X 5 X 5 X 5 X 5 X 5

Hey guys! Today, we're diving into a neat little math problem. We need to figure out which expression is the same as $3 ims 5 ims 5 ims 5 ims 5 ims 5 ims 5$. Sounds like fun, right? Let's break it down step by step so it's super easy to understand. Our goal is to simplify the given expression and match it with one of the options provided. We'll look at how exponents work and how to manipulate them to get the correct answer. By the end of this, you'll be a pro at simplifying expressions like this! So, grab your thinking caps, and let's get started!

Understanding the Problem

Before we jump into solving, let's make sure we all understand what the problem is asking. We have the expression $3 ims 5 ims 5 ims 5 ims 5 ims 5 ims 5$. What this means is that we're multiplying 3 by 5, and then we're multiplying that result by 5 again, and so on, a total of six times. Essentially, we need to rewrite this expression in a more compact form using exponents. Exponents are a handy way to show repeated multiplication. For example, $5 ims 5 ims 5$ can be written as $5^3$. So, our mission is to rewrite the given expression using exponents and then compare it to the answer choices. This involves recognizing the base and the exponent in the repeated multiplication. Once we do that, we can easily see which of the given options matches our simplified expression. Remember, the key is to accurately count how many times 5 is being multiplied by itself and then express that as an exponent. This will make the problem much easier to solve!

Simplifying the Expression

Okay, let's simplify the expression $3 ims 5 ims 5 ims 5 ims 5 ims 5 ims 5$. The first thing we notice is that the number 5 is being multiplied by itself several times. To be exact, 5 is being multiplied by itself six times. We can rewrite this repeated multiplication using an exponent. An exponent tells us how many times a number (the base) is multiplied by itself. In this case, the base is 5, and it's being multiplied by itself six times. So, we can write $5 ims 5 ims 5 ims 5 ims 5 ims 5$ as $5^6$. Now, let's bring back the 3 that's being multiplied at the beginning of the expression. So, our simplified expression becomes $3 ims 5^6$. This means we're multiplying 3 by 5 raised to the power of 6. This is a much more compact and easier-to-understand way of writing the original expression. The key here was to recognize the repeated multiplication of 5 and express it using an exponent. Now that we've simplified the expression, we can easily compare it to the answer choices and find the one that matches.

Evaluating the Answer Choices

Now that we've simplified the original expression to $3 ims 5^6$, let's take a look at the answer choices and see which one matches. Here are the options:

A. $15^6$ B. $15^7$ C. $3(5)^6$ D. $3(5)^7$

Let's go through each option:

• Option A: $15^6$

  This means $15 ims 15 ims 15 ims 15 ims 15 ims 15$. This is not the same as $3 ims 5^6$ because $15^6$ implies that 15 is being multiplied by itself six times, whereas our expression has 3 multiplied by 5 raised to the power of 6. These are fundamentally different.

• Option B: $15^7$

  Similarly, $15^7$ means $15 ims 15 ims 15 ims 15 ims 15 ims 15 ims 15$, which is also not the same as $3 ims 5^6$. This option raises 15 to the power of 7, which is not what our simplified expression represents.

• Option C: $3(5)^6$

  This is exactly the same as $3 ims 5^6$. The parentheses around the 5 indicate that 5 is raised to the power of 6, and then that result is multiplied by 3. This perfectly matches our simplified expression.

• Option D: $3(5)^7$

  This means $3 ims 5^7$, which is not the same as $3 ims 5^6$. Here, 5 is raised to the power of 7, but in our expression, 5 is raised to the power of 6.

So, after evaluating each option, we can clearly see that option C, $3(5)^6$, is the only one that is equivalent to our simplified expression.

The Correct Answer

Alright, guys, after simplifying the expression and evaluating the answer choices, we've found the correct answer! The expression equivalent to $3 ims 5 ims 5 ims 5 ims 5 ims 5 ims 5$ is:

C. $3(5)^6$

This means that when we multiply 3 by 5 raised to the power of 6, we get the same result as multiplying 3 by 5 six times. This exercise shows how exponents can simplify expressions and make them easier to work with. Remember, the key is to accurately identify the base and the exponent in the repeated multiplication. Once you do that, you can easily rewrite the expression in a more compact form. And that's it! We've successfully solved the problem and found the equivalent expression. Great job, everyone!

Tips for Similar Problems

When you encounter similar problems, here are a few tips to keep in mind:

1. Identify Repeated Multiplication:

   Look for numbers that are being multiplied by themselves multiple times. This is a clear indicator that you can use exponents to simplify the expression. For instance, in the expression $2 ims 2 ims 2 ims 2$, you can see that 2 is being multiplied by itself four times. This can be written as $2^4$.

2. Count the Number of Multiplications:

   Carefully count how many times the number is being multiplied by itself. This number will be the exponent. In the example above, 2 is multiplied by itself four times, so the exponent is 4.

3. Rewrite Using Exponents:

   Rewrite the repeated multiplication using exponents. The base is the number being multiplied, and the exponent is the number of times it's being multiplied. So, $2 ims 2 ims 2 ims 2$ becomes $2^4$.

4. Pay Attention to Other Factors:

   Don't forget to include any other numbers that are being multiplied in the expression. For example, if the expression is $3 ims 2 ims 2 ims 2 ims 2$, you would rewrite the repeated multiplication of 2 as $2^4$ and then multiply it by 3, resulting in $3 ims 2^4$.

5. Simplify Before Comparing:

   Always simplify the expression as much as possible before comparing it to the answer choices. This will make it easier to identify the correct answer.

6. Understand Exponent Rules:

   Familiarize yourself with basic exponent rules, such as $a^m ims a^n = a^{m+n}$ and $(a^m)^n = a^{mn}$. These rules can help you simplify more complex expressions.

7. Practice Regularly:

   The more you practice, the more comfortable you'll become with simplifying expressions using exponents. Try solving a variety of problems to build your skills and confidence.

By following these tips, you'll be well-equipped to tackle similar problems and simplify expressions with ease. Remember, the key is to break down the problem into smaller, manageable steps and to understand the basic principles of exponents.