Simplifying Exponential Expressions: A Step-by-Step Guide

Dec 3, 2025 by Alex Johnson 58 views

Have you ever stared at an equation filled with exponents and felt a little lost? Don't worry, you're not alone! Exponents can seem tricky at first, but with a few key rules, you can simplify even the most complex expressions. In this guide, we'll break down how to simplify the expression (7^(12/5))(7(-2/5)) using the laws of exponents. We'll walk through each step in detail, so you'll not only get the answer but also understand the why behind it. This understanding is what truly empowers you to tackle any exponent problem that comes your way. We will explore the fundamental laws that govern how exponents behave, especially when dealing with multiplication of terms that share a common base. This foundational knowledge will enable you to confidently approach and solve similar problems in the future, making complex mathematical expressions much more manageable. So, let’s dive in and unlock the secrets of exponents together! By the end of this article, you’ll be equipped with the knowledge and confidence to tackle similar problems on your own.

Understanding the Laws of Exponents

Before we jump into solving the specific expression, let's quickly review the fundamental laws of exponents. These rules are the building blocks for simplifying any exponential expression, and grasping them is crucial. Exponents represent repeated multiplication. For example, x^3 means x * x * x. The base (x in this case) is the number being multiplied, and the exponent (3) tells you how many times to multiply the base by itself. Now, let’s consider the laws that are most relevant to our problem:

Product of Powers Rule: This rule states that when you multiply two exponents with the same base, you add the powers. Mathematically, it's expressed as: a^m * a^n = a^(m+n). This is the primary rule we will be using to solve our problem. Think of it this way: If you have 'm' number of 'a's multiplied together and you multiply that by 'n' number of 'a's, then you simply have 'm+n' number of 'a's multiplied together. Understanding this intuitively can help you remember the rule better.
Negative Exponent Rule: A negative exponent indicates a reciprocal. Specifically, a^(-n) = 1/a^n. This means that any base raised to a negative power is equal to its reciprocal raised to the positive version of that power. This rule is incredibly useful for simplifying expressions and eliminating negative exponents.
Fractional Exponents: Fractional exponents represent roots. For instance, a^(1/n) is the nth root of a. So, a^(1/2) is the square root of a, a^(1/3) is the cube root of a, and so on. This connection between fractional exponents and roots is vital for simplifying expressions involving radicals.

With these rules in mind, we are well-equipped to tackle our expression. The key takeaway here is the Product of Powers Rule, as it directly applies to our problem. Remember, the goal is always to simplify the expression by combining like terms and reducing it to its most basic form. Mastering these exponent rules not only helps in solving specific problems but also builds a strong foundation for more advanced mathematical concepts.

Applying the Product of Powers Rule

Now, let's get to the heart of the problem: simplifying (7^(12/5))(7(-2/5)). The most important thing to notice here is that we are multiplying two exponential terms, and both terms have the same base, which is 7. This is a perfect scenario for applying the Product of Powers Rule that we discussed earlier. Remember, this rule states that when multiplying exponents with the same base, we add the powers. In our case, the base is 7, and the powers are 12/5 and -2/5.

So, according to the Product of Powers Rule, we can rewrite the expression as: 7^((12/5) + (-2/5)). The next step is to simply add the exponents. We have two fractions, 12/5 and -2/5, which share a common denominator, making the addition straightforward. Adding the numerators, we get 12 + (-2) = 10. Therefore, the sum of the exponents is 10/5.

Our expression now looks like this: 7^(10/5). We're not quite done yet, as we can further simplify the exponent. The fraction 10/5 can be reduced to its simplest form. Dividing both the numerator and the denominator by their greatest common divisor, which is 5, we get 10/5 = 2. This simplification is crucial because it allows us to express the exponent in its most basic form, making the final calculation easier.

By applying the Product of Powers Rule and simplifying the resulting exponent, we've transformed the original expression into a much more manageable form. This step-by-step approach highlights the power of understanding and applying the fundamental laws of exponents. Remember, the key is to break down complex problems into smaller, more manageable steps. This not only makes the problem less daunting but also ensures accuracy in your calculations.

Simplifying the Exponent

As we saw in the previous section, after applying the Product of Powers Rule, our expression became 7^(10/5). Now, we need to simplify the exponent 10/5. This is a straightforward fraction that can be easily reduced. The fraction 10/5 represents 10 divided by 5, which equals 2. Therefore, we can simplify the exponent 10/5 to 2.

This simplification is a critical step in solving the problem. It transforms the expression from 7^(10/5) to 7^2, which is much easier to evaluate. Simplifying exponents whenever possible is a best practice in mathematics. It not only makes the calculations simpler but also helps in understanding the true value of the expression. In this case, reducing the fraction allows us to see that the original complex-looking exponent actually represents a simple whole number.

The importance of simplifying exponents cannot be overstated. It's a fundamental skill that applies to a wide range of mathematical problems, from basic algebra to more advanced calculus. By always looking for opportunities to simplify, you can make your work cleaner, more efficient, and less prone to errors. Moreover, simplifying often reveals hidden patterns and relationships within the expression, providing a deeper understanding of the underlying mathematical concepts. In our example, simplifying the exponent made it immediately clear that we were dealing with a simple square of 7, which brings us to our final step: evaluating the expression.

Evaluating the Expression

After simplifying the exponent, our expression is now 7^2. This is a simple exponential expression that is easy to evaluate. Remember, an exponent indicates repeated multiplication. In this case, 7^2 means 7 multiplied by itself, or 7 * 7.

Performing this multiplication, we find that 7 * 7 = 49. Therefore, the simplified value of the expression (7^(12/5))(7(-2/5)) is 49. This is our final answer. We have successfully simplified the original expression by applying the Product of Powers Rule, simplifying the exponent, and then evaluating the result.

This final step demonstrates the culmination of all our efforts. We started with a seemingly complex expression involving fractional and negative exponents, and through a series of logical steps, we reduced it to a simple whole number. This process highlights the power of the laws of exponents in making complex mathematical problems manageable. The ability to evaluate such expressions is a fundamental skill in mathematics and is crucial for various applications in science, engineering, and other fields.

By walking through each step methodically, we not only arrived at the solution but also gained a deeper understanding of the underlying principles. This understanding is what truly empowers you to solve similar problems on your own. Remember, mathematics is not just about finding the right answer; it's about understanding the process and the reasoning behind it.

Conclusion

In this guide, we walked through the process of simplifying the expression (7^(12/5))(7(-2/5)) using the laws of exponents. We started by understanding the fundamental rules, particularly the Product of Powers Rule, which states that when multiplying exponents with the same base, you add the powers. We then applied this rule to our expression, simplified the resulting exponent, and finally, evaluated the expression to arrive at the answer: 49. This step-by-step approach not only solved the problem but also provided a clear understanding of the underlying concepts. Mastering these techniques will empower you to confidently tackle similar problems involving exponents.

The key takeaways from this exercise are the importance of understanding the laws of exponents, the value of simplifying expressions whenever possible, and the power of breaking down complex problems into smaller, manageable steps. Remember to always look for opportunities to apply the Product of Powers Rule when multiplying terms with the same base. Simplifying exponents, especially fractional ones, can significantly ease the calculations. And most importantly, practice makes perfect. The more you work with exponents, the more comfortable and confident you will become.

We encourage you to explore more complex problems and continue practicing these skills. The world of exponents is vast and fascinating, and the more you delve into it, the more you will appreciate its power and elegance. For further learning and practice, you might find the resources at Khan Academy's Exponents and Radicals section helpful. Keep exploring, keep practicing, and you'll become an exponent expert in no time!