Correct Equality: $\sqrt[7]{x^{-2}}$ Options & Solution

Hey guys! Let's break down this radical expression and find the correct equality. We're dealing with some exponents and roots here, so let's make sure we understand the rules before diving into the options.

Understanding the Basics

Before we tackle the problem, let's quickly review the fundamental relationship between radicals and exponents. Remember, a radical expression like $\sqrt[n]{a^m}$ can be rewritten using fractional exponents. The general rule is:

$\sqrt[n]{a^m} = a^{\frac{m}{n}}$

Where:

• n is the index of the radical (the small number indicating the root, like the 7 in our problem).
• a is the base (the expression under the radical, in our case, x).
• m is the exponent of the base (the power to which x is raised).

This rule is super important because it allows us to convert between radical form and exponential form, which can make simplifying expressions much easier. Think of the index n as the denominator of the fractional exponent, and the exponent m inside the radical as the numerator. Now, let’s also refresh our minds on negative exponents. A negative exponent indicates a reciprocal. In other words:

$x^{-n} = \frac{1}{x^n}$

This means that $x^{-2}$ is the same as $\frac{1}{x^2}$. Keeping these two rules in mind – the conversion between radicals and fractional exponents, and the meaning of negative exponents – will help us solve the given problem and similar algebraic challenges. It's like having the right tools in your toolbox for any math problem that comes your way!

Applying the Rules to Our Problem

Now, let's apply these rules to the expression in the question: $\sqrt[7]{x^{-2}}$.

1. Identify the components: In this expression, the index n is 7, the base a is x, and the exponent m is -2. Remember, that negative exponent is key!
2. Convert to fractional exponent: Using the rule $\sqrt[n]{a^m} = a^{\frac{m}{n}}$, we can rewrite $\sqrt[7]{x^{-2}}$ as $x^{\frac{-2}{7}}$.

And that's it! We've successfully converted the radical expression into an equivalent expression with a fractional exponent. This is a crucial step because it allows us to directly compare our result with the given options. So, by understanding the basic rules of radicals and exponents, especially how to convert between radical and exponential forms, we've made the problem much simpler to solve. It’s all about breaking down the expression into manageable parts and applying the relevant rules step by step.

Analyzing the Options

Okay, we've figured out that $\sqrt[7]{x^{-2}}$ is the same as $x^{\frac{-2}{7}}$. Now, let's take a look at the options and see which one matches our result:

a) $\sqrt[7]{x^{-2}} = x^{\frac{2}{-7}}$ b) $\sqrt[7]{x^{-2}} = x^{\frac{-2}{7}}$ c) $\sqrt[7]{x^{-2}} = x^{7}$ d) $\sqrt[7]{x^{-2}} = x^{\frac{-7}{2}}$

Let's go through them one by one:

• Option a: $x^{\frac{2}{-7}}$. This is the same as $x^{\frac{-2}{7}}$ because a fraction with a negative denominator is equivalent to a fraction with a negative numerator. So, this option looks promising!
• Option b: $x^{\frac{-2}{7}}$. This is exactly what we got when we converted the original expression! This is definitely a correct answer.
• Option c: $x^{7}$. This is way off. We know the exponent should be a fraction, not a whole number. So, we can eliminate this option right away.
• Option d: $x^{\frac{-7}{2}}$. This is also incorrect. The numerator and denominator are flipped compared to what we calculated. So, this one's out too.

The Correct Answer

After carefully analyzing the options, we can see that option b is the correct equality: $\sqrt[7]{x^{-2}} = x^{\frac{-2}{7}}$. And, just to be super clear, option a is also correct because $\frac{2}{-7}$ is the same as $\frac{-2}{7}$. Both represent the same value. So, in this case, there might be a slight ambiguity in the question, as both a) and b) are mathematically equivalent. However, the most direct and simplified form of our result matches option b perfectly.

Common Mistakes to Avoid

When dealing with radicals and exponents, it's easy to make a few common mistakes. Let's highlight them so you can steer clear of these pitfalls:

1. Forgetting the negative sign: When you have a negative exponent inside the radical, like in our problem with $x^{-2}$, don't forget to carry that negative sign over when you convert to a fractional exponent. It's a small detail, but it makes a big difference in the final answer.
2. Flipping the fraction: Remember, the index of the radical (the little number outside the radical symbol) becomes the denominator of the fractional exponent, and the exponent of the base inside the radical becomes the numerator. Don't flip them around!
3. Misunderstanding negative exponents: A negative exponent doesn't mean the number becomes negative. It means you take the reciprocal of the base raised to the positive exponent. For example, $x^{-2}$ is $\frac{1}{x^2}$, not -x². Getting this wrong is a common mistake, so always double-check your understanding.
4. Not simplifying: Sometimes, you might arrive at an answer that's technically correct but not in its simplest form. Always try to simplify your answer as much as possible. In our case, recognizing that $\frac{2}{-7}$ is the same as $\frac{-2}{7}$ helps you choose the most simplified option.

By being aware of these common mistakes, you can avoid them and boost your confidence in solving problems involving radicals and exponents. Math is all about precision, so paying attention to these details will definitely pay off!

Practice Problems

Want to make sure you've really got this down? Here are a few practice problems you can try:

1. Rewrite $\sqrt[5]{x^3}$ using fractional exponents.
2. Simplify $\sqrt[4]{x^{-8}}$ and express it with a positive exponent.
3. Which of the following is equivalent to $x^{\frac{3}{4}}$? a) $\sqrt[3]{x^4}$ b) $\sqrt[4]{x^3}$ c) $(\sqrt[3]{x})^4$ d) $(\sqrt[4]{x})^3$

Working through these problems will help solidify your understanding of the concepts we've discussed. The more you practice, the more comfortable you'll become with radicals and exponents. Remember, math is like any skill – the more you use it, the better you get at it. So, grab a pencil and paper and give these a try!

Solutions to Practice Problems

Let's check the solutions to the practice problems:

1. $\sqrt[5]{x^3} = x^{\frac{3}{5}}$
2. $\sqrt[4]{x^{-8}} = x^{\frac{-8}{4}} = x^{-2} = \frac{1}{x^2}$
3. The correct answer is b) $\sqrt[4]{x^3}$ and d) $(\sqrt[4]{x})^3$. Both are equivalent to $x^{\frac{3}{4}}$. Remember that $a^{\frac{m}{n}}$ can be interpreted as both $\sqrt[n]{a^m}$ and $(\sqrt[n]{a})^m$.

How did you do? If you got them all right, awesome! You're well on your way to mastering radicals and exponents. If you struggled with any of them, don't worry. Go back and review the concepts and examples we discussed. Identify where you went wrong and try the problem again. The key is to learn from your mistakes and keep practicing.

Wrapping Up

So, guys, we've tackled a tricky little problem involving radicals and exponents. We figured out that the correct equality is $\sqrt[7]{x^{-2}} = x^{\frac{-2}{7}}$ (or $x^{\frac{2}{-7}}$, which is the same thing!). We also talked about the important rules to remember, common mistakes to avoid, and even gave you some practice problems to try out.

The key takeaway here is that understanding the relationship between radicals and fractional exponents is crucial for simplifying these kinds of expressions. And, of course, paying attention to those sneaky negative signs is super important!

Keep practicing, and you'll be a pro at these in no time. You got this!