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

Hey guys! Let's dive into a math problem that might seem a bit tricky at first, but I promise it's totally manageable once we break it down. We're going to evaluate an expression involving fractional exponents. Specifically, we need to find the value of: $\left(8^{1 / 4}\right) \cdot \left(8^{1 / 4}\right) \cdot \left(8^{1 / 4}\right) \cdot \left(8^{1 / 4}\right)$. This looks intimidating, but trust me, it's easier than it looks!

Understanding Fractional Exponents

Before we jump into solving the problem, let's quickly recap what fractional exponents actually mean. A fractional exponent like $a^{m/n}$ can be interpreted in two ways: as the $n$-th root of $a$ raised to the power of $m$, or as $a$ raised to the power of $m$ and then taking the $n$-th root. Mathematically, this is represented as $a^{m/n} = (\sqrt[n]{a})^m = \sqrt[n]{a^m}$. Understanding this equivalence is key to simplifying expressions with fractional exponents. Fractional exponents are not as scary as they seem; they're just a different way of expressing roots and powers. Think of $8^{1/4}$ as the fourth root of 8. When you see a fractional exponent, remember that the denominator tells you what root to take, and the numerator tells you what power to raise the base to. So, $x^{1/2}$ is the square root of x, $x^{1/3}$ is the cube root of x, and so on. Keep this in mind as we tackle the problem.

Breaking Down the Problem

Okay, now that we've refreshed our understanding of fractional exponents, let's get back to the expression: $\left(8^{1 / 4}\right) \cdot \left(8^{1 / 4}\right) \cdot \left(8^{1 / 4}\right) \cdot \left(8^{1 / 4}\right)$. Notice that we are multiplying the same base ($8^{1/4}$) by itself four times. A neat trick we can use here is the rule of exponents that states when you multiply numbers with the same base, you add their exponents. In other words, $a^m \cdot a^n = a^{m+n}$. Applying this rule to our expression, we get:

$8^{1/4} \cdot 8^{1/4} \cdot 8^{1/4} \cdot 8^{1/4} = 8^{(1/4 + 1/4 + 1/4 + 1/4)} = 8^{4/4} = 8^1 = 8$

So, the expression simplifies to $8^1$, which is simply 8. Isn't that neat? By understanding and applying the basic rules of exponents, we were able to transform a seemingly complex expression into a straightforward calculation. Remember, math is all about recognizing patterns and applying the right tools. The beauty of math lies in its ability to simplify complex problems into manageable steps. By breaking down the expression and using the properties of exponents, we transformed a complex-looking product into a simple power of 8, making the final calculation straightforward and easy to understand. With practice, you'll start to see these patterns more quickly and confidently.

Alternative Approach

Another way to think about this problem is to recognize that we are essentially raising $8^{1/4}$ to the fourth power. That is, $\left(8^{1 / 4}\right) \cdot \left(8^{1 / 4}\right) \cdot \left(8^{1 / 4}\right) \cdot \left(8^{1 / 4}\right) = \left(8^{1 / 4}\right)^4$. Using the property of exponents that $(a^m)^n = a^{m \cdot n}$, we have:

$\left(8^{1 / 4}\right)^4 = 8^{(1/4) \cdot 4} = 8^1 = 8$

Again, we arrive at the same answer: 8. This approach highlights another important rule of exponents and provides a different perspective on how to simplify the expression. The power of a power rule is a fundamental concept in algebra, allowing us to simplify expressions involving nested exponents. This approach demonstrates how applying different exponent rules can lead to the same simplified result. This flexibility is a valuable asset in mathematics, as it allows you to choose the method that best suits your understanding and the problem at hand. By recognizing that the given expression is equivalent to raising $8^{1/4}$ to the fourth power, we can quickly apply the power of a power rule and simplify the expression in a single step.

Why the Other Options Are Wrong

Let's quickly address why the other answer choices are incorrect:

• A. 64: This would be the answer if we were multiplying 8 by itself four times (i.e., $8^4$), not if we were dealing with the fourth root. This is a common mistake, so always double-check what the exponent is telling you to do.
• C. $2 \sqrt{8}$: This is equivalent to $2 \cdot 2 \sqrt{2} = 4 \sqrt{2}$, which doesn't match our simplified expression of 8.
• D. $\sqrt{8}$: This would be $8^{1/2}$, which is the square root of 8, not the result of multiplying the fourth root of 8 by itself four times. Remember that the key to solving these problems is understanding what the fractional exponents represent and applying the correct exponent rules.

Final Answer

Therefore, the correct answer is B. 8. I hope this explanation helps you guys understand how to simplify expressions with fractional exponents! Keep practicing, and you'll become a pro in no time. Remember, the world of exponents and roots is full of patterns and shortcuts. Once you master these fundamental rules, you'll be equipped to tackle a wide range of mathematical challenges with confidence and ease. So, keep exploring, keep practicing, and never stop learning!