Simplifying Expressions With Integer Exponents: A Step-by-Step Guide

Oct 30, 2025 by ADMIN 69 views

Hey guys! Let's dive into the fascinating world of integer exponents and simplify some expressions. We're going to break down a specific problem step-by-step, making sure everyone understands the logic behind each move. So, grab your pencils, and let's get started!

Understanding the Question

The question asks us to find an equivalent expression for $\frac{\left(15^{-3} ullet 4^7\right)^0}{4^{-3}}$ using the properties of integer exponents. In simpler terms, we need to take this seemingly complicated expression and simplify it into something more manageable. Don't worry, it's not as daunting as it looks! We'll use the rules of exponents to make it easy. The question highlights the importance of knowing and applying these properties correctly. Before we jump into solving the problem, let's quickly recap the key exponent rules that we'll be using. Remember, mastering these rules is crucial for simplifying any expression involving exponents. So, let's refresh our memory and then tackle the problem head-on!

Key Properties of Integer Exponents

Before we tackle the problem, let's quickly recap some key properties of integer exponents that will come in handy:

Zero Exponent: Any non-zero number raised to the power of 0 is 1. That is, $a^0 = 1$ (where $a \neq 0$ ).
Negative Exponent: A number raised to a negative exponent is equal to the reciprocal of that number raised to the positive exponent. That is, $a^{-n} = \frac{1}{a^n}$ .
Power of a Product: The power of a product is the product of the powers. That is, $(ab)^n = a^n b^n$ .
Power of a Quotient: The power of a quotient is the quotient of the powers. That is, $(\frac{a}{b})^n = \frac{a^n}{b^n}$ .
Product of Powers: When multiplying powers with the same base, add the exponents. That is, $a^m ullet a^n = a^{m+n}$ .
Quotient of Powers: When dividing powers with the same base, subtract the exponents. That is, $\frac{a^m}{a^n} = a^{m-n}$ .
Power of a Power: When raising a power to another power, multiply the exponents. That is, $(a^m)^n = a^{mn}$ .

With these properties in mind, we're well-equipped to simplify the given expression. These rules are the foundation for working with exponents, and understanding them deeply will help you tackle a wide range of problems. So, keep these in your toolbox, and let's move on to applying them to our specific question!

Solving the Expression Step-by-Step

Now, let's apply these properties of exponents to simplify the expression $\frac{\left(15^{-3} ullet 4^7\right)^0}{4^{-3}}$ . We'll take it one step at a time to make sure we don't miss anything.

Step 1: Dealing with the Zero Exponent

The first thing we notice is the term $\left(15^{-3} ullet 4^7\right)^0$ . Remember the zero exponent rule? Any non-zero number raised to the power of 0 is 1. So, this entire term simplifies to 1. This is a crucial first step, as it significantly simplifies the expression. Recognizing and applying the zero exponent rule is often the key to unlocking more complex problems. It's like finding the easy way out of a maze – once you spot it, the rest becomes much simpler. So, let's replace that term with 1 and see what we have now.

Our expression now looks like this: $\frac{1}{4^{-3}}$ .

Step 2: Handling the Negative Exponent

Next up, we have $4^{-3}$ in the denominator. This is where the negative exponent rule comes into play. Recall that $a^{-n} = \frac{1}{a^n}$ . So, $4^{-3}$ is the same as $\frac{1}{4^3}$ . Applying the negative exponent rule allows us to rewrite terms with negative exponents as fractions, which often makes further simplification easier. It's like flipping a fraction to make it more manageable. So, let's apply this rule and see what our expression looks like.

We can rewrite $\frac{1}{4^{-3}}$ as $\frac{1}{\frac{1}{4^3}}$ .

Step 3: Simplifying the Fraction

Now we have a fraction within a fraction, which might look a bit intimidating, but it's actually quite simple to handle. Dividing by a fraction is the same as multiplying by its reciprocal. So, dividing 1 by $\frac{1}{4^3}$ is the same as multiplying 1 by $4^3$ . This is a fundamental concept in fraction manipulation, and it's essential for simplifying complex expressions. It's like turning a division problem into a multiplication problem – much easier to handle! So, let's make that transformation.

This simplifies to $1 ullet 4^3$ , which is just $4^3$ .

Step 4: Calculating the Final Value

Finally, we need to calculate $4^3$ . This means 4 multiplied by itself three times: $4 ullet 4 ullet 4$ . Let's do the math.

$4^3 = 4 ullet 4 ullet 4 = 16 ullet 4 = 64$

So, our simplified expression is 64!

Matching with the Given Options

Now that we've simplified the expression, let's see which of the given options matches our result:

A. $1 ullet 4^{-3}$ B. $\frac{60}{4^{-3}}$ C. $\frac{1}{64}$ D. 64

The correct answer is D. 64. We've successfully simplified the original expression using the properties of integer exponents and arrived at the correct answer. Great job, guys! You've navigated through the steps with precision and now have a solid understanding of how to tackle similar problems.

Why Other Options are Incorrect

It's just as important to understand why the other options are incorrect as it is to know the correct answer. This helps solidify your understanding of the concepts and prevent similar mistakes in the future. So, let's take a quick look at why options A, B, and C are not the correct answers.

A. $1 ullet 4^{-3}$ : This option represents the expression after applying the zero exponent rule but before handling the negative exponent. It's an intermediate step, not the final simplified form. Choosing this option might indicate a misunderstanding of the need to fully simplify the expression.
B. $\frac{60}{4^{-3}}$ : This option seems to introduce an arbitrary number (60) and doesn't follow the correct application of exponent rules. It's a clear deviation from the proper simplification process and likely a result of a misunderstanding of the fundamental principles.
C. $\frac{1}{64}$ : This option is the reciprocal of the correct answer. It might arise from incorrectly applying the negative exponent rule or confusing the final result with an intermediate step involving a fraction. It highlights the importance of carefully tracking each step and ensuring the correct operation is performed.

Understanding why these options are wrong reinforces your grasp of the concepts and helps you avoid common pitfalls. So, keep this in mind as you tackle similar problems in the future!

Conclusion

Simplifying expressions with integer exponents might seem tricky at first, but by understanding and applying the key properties, you can break down even the most complex problems. Remember the zero exponent, negative exponent, and other rules we discussed. Practice makes perfect, so keep working on these types of problems, and you'll become a pro in no time! Keep practicing, and you'll find these problems becoming second nature. And remember, we're here to help you along the way. So, keep those questions coming, and let's keep learning together! You've got this!