Equivalent Expression: Simplifying Radicals And Exponents

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Equivalent Expression: Simplifying Radicals and Exponents

Let's break down how to find the equivalent expression for the given radical expression. This involves understanding the rules of exponents and how they relate to radicals. It might seem tricky at first, but once you grasp the core concepts, you'll be simplifying expressions like a pro, guys!

Understanding the Problem

We're given the expression \sqrt[16]{\frac{1}{x^5 ullet x^3}} and a few options to choose from. Our mission, should we choose to accept it, is to simplify the given expression and see which of the options matches. The key here is to manipulate the expression using exponent rules until we arrive at a simpler form. So, let's roll up our sleeves and get started!

Before we dive into the solution, let's quickly recap some essential exponent rules:

  1. Product of powers: x^m ullet x^n = x^{m+n}
  2. Quotient of powers: xmxn=xmβˆ’n\frac{x^m}{x^n} = x^{m-n}
  3. Power of a power: (x^m)^n = x^{m ullet n}
  4. Negative exponent: xβˆ’n=1xnx^{-n} = \frac{1}{x^n}
  5. Fractional exponent: xmn=xmn\sqrt[n]{x^m} = x^{\frac{m}{n}}

These rules are our trusty tools in simplifying expressions. Make sure you have them handy as we work through the problem.

Step-by-Step Solution

Let's tackle this step by step, making sure we understand each transformation. It's like building with LEGOs – each step adds to the final masterpiece.

Step 1: Simplify the denominator inside the radical

We start with \sqrt[16]{\frac{1}{x^5 ullet x^3}}. Notice that we have x5x^5 multiplied by x3x^3 in the denominator. Using the product of powers rule, we can combine these:

x^5 ullet x^3 = x^{5+3} = x^8

So, our expression now becomes:

1x816\sqrt[16]{\frac{1}{x^8}}

Step 2: Rewrite the fraction using a negative exponent

Now, we have a fraction inside the radical. To make things easier, let's use the negative exponent rule to rewrite 1x8\frac{1}{x^8} as xβˆ’8x^{-8}. Remember, a negative exponent means we're dealing with the reciprocal of the base raised to the positive exponent. Our expression transforms to:

xβˆ’816\sqrt[16]{x^{-8}}

Step 3: Convert the radical to a fractional exponent

This is where the fractional exponent rule comes into play. Recall that xmn=xmn\sqrt[n]{x^m} = x^{\frac{m}{n}}. In our case, we have the 16th root of xβˆ’8x^{-8}, so we can rewrite it as:

xβˆ’816x^{-\frac{8}{16}}

Step 4: Simplify the fractional exponent

We're almost there! Now, let's simplify the fraction βˆ’816-\frac{8}{16}. Both 8 and 16 are divisible by 8, so we can reduce the fraction:

βˆ’816=βˆ’12-\frac{8}{16} = -\frac{1}{2}

Thus, our expression simplifies to:

xβˆ’12x^{-\frac{1}{2}}

Step 5: Match the simplified expression with the options

Looking at the options provided, we see that our simplified expression, xβˆ’12x^{-\frac{1}{2}}, matches option B. So, the equivalent expression is indeed xβˆ’12x^{-\frac{1}{2}}.

Why Other Options Are Incorrect

It's always good to understand why the other options are incorrect. This helps solidify our understanding of the concepts.

  • Option A (xβˆ’2x^{-2}): This would be the answer if we had simplified the exponent to -2, which isn't the case here. We correctly simplified it to -1/2.
  • Option C (x2x^2): This is the opposite sign and a different exponent value. This would only be correct if we somehow ended up with a positive 2 in the exponent, which didn't happen in our steps.
  • Option D (x12x^{\frac{1}{2}}): This has the correct exponent value (1/2) but the wrong sign. We ended up with a negative exponent because of the initial fraction inside the radical.

Key Takeaways

  • Master the exponent rules: These rules are your best friends when simplifying expressions involving exponents and radicals. Memorize them and practice using them.
  • Break it down: Complex expressions can be intimidating, but breaking them down into smaller, manageable steps makes the process much easier.
  • Simplify fractions: Always simplify fractional exponents to their simplest form. This makes it easier to compare expressions and choose the correct answer.
  • Double-check your work: It's easy to make small mistakes, especially with negative signs. Always double-check each step to ensure accuracy.

Additional Practice

To really nail this concept, try simplifying similar expressions. Here are a few practice problems:

  1. \sqrt[9]{\frac{1}{x^2 ullet x}}
  2. xβˆ’105\sqrt[5]{x^{-10}}
  3. 1x4\frac{1}{\sqrt{x^4}}

Work through these problems step-by-step, and you'll become a pro at simplifying radical and exponential expressions in no time. You've got this, guys!

Conclusion

Simplifying expressions with radicals and exponents might seem daunting initially, but by systematically applying the rules of exponents and breaking down the problem into smaller steps, we can arrive at the correct answer. In this case, the equivalent expression for \sqrt[16]{\frac{1}{x^5 ullet x^3}} is xβˆ’12x^{-\frac{1}{2}}. Remember to practice regularly, and you'll become more confident in tackling these types of problems. Keep up the great work!