Simplifying Radicals: $(\sqrt[4]{3})^7$ Explained

Hey guys! Let's dive into a common problem you might encounter in mathematics: simplifying expressions with radicals and exponents. Today, we're going to break down the expression $(\sqrt[4]{3})^7$ and figure out which of the provided options is equivalent. This type of problem tests your understanding of how radicals and exponents interact, and mastering it can give you a solid edge in algebra and beyond. So, grab your thinking caps, and let’s get started!

Understanding the Basics

Before we tackle the main problem, let's quickly review the basic principles that govern radicals and exponents. These fundamentals are crucial for simplifying complex expressions and ensuring you don't get lost in the mathematical weeds. It's like knowing the rules of the road before you start driving – essential for a smooth journey!

Radicals

A radical, often represented by the symbol $\sqrt[n]{x}$, indicates the $n$-th root of $x$. The number $n$ is called the index of the radical, and $x$ is the radicand. For example, $\sqrt[2]{9}$ (commonly written as $\sqrt{9}$) is the square root of 9, which is 3, because $3^2 = 9$. Similarly, $\sqrt[3]{8}$ is the cube root of 8, which is 2, because $2^3 = 8$. Understanding this notation is the first step in simplifying expressions with radicals.

Exponents

An exponent indicates how many times a number (the base) is multiplied by itself. For instance, $x^n$ means $x$ multiplied by itself $n$ times. So, $2^3 = 2 \times 2 \times 2 = 8$. Exponents can also be fractions, which brings us to the connection between exponents and radicals. A fractional exponent like $x^{a/b}$ is equivalent to taking the $b$-th root of $x$ and then raising it to the power of $a$. Mathematically, this is represented as $x^{a/b} = \sqrt[b]{x^a} = (\sqrt[b]{x})^a$. This relationship is key to converting between radical and exponential forms and is super handy for simplifying expressions.

The Connection

The link between radicals and exponents is vital for simplifying expressions. The $n$-th root of $x$, written as $\sqrt[n]{x}$, can be expressed as $x^{1/n}$. This means that $\sqrt{x} = x^{1/2}$, $\sqrt[3]{x} = x^{1/3}$, and so on. This conversion allows us to use the rules of exponents to manipulate and simplify expressions involving radicals. For instance, if you have $\sqrt[4]{16}$, you can rewrite it as $16^{1/4}$, which equals 2 because $2^4 = 16$. Remembering this connection will make simplifying radicals much easier.

Solving the Problem: $(\sqrt[4]{3})^7$

Now that we've refreshed our understanding of radicals and exponents, let's tackle the given expression: $(\sqrt[4]{3})^7$. Our goal is to rewrite this expression in the form of a base raised to a single exponent. Here’s how we can do it step by step:

1. Convert the radical to exponential form:

   The expression $\sqrt[4]{3}$ can be rewritten using a fractional exponent. Remember that $\sqrt[n]{x} = x^{1/n}$. Therefore, $\sqrt[4]{3} = 3^{1/4}$. This conversion is the cornerstone of simplifying the expression. By changing the radical to an exponent, we can apply the rules of exponents more easily.

2. Substitute and simplify:

   Now, substitute $3^{1/4}$ back into the original expression: $(\sqrt[4]{3})^7 = (3^{1/4})^7$. Here, we have an exponent raised to another exponent. According to the power of a power rule, $(x^a)^b = x^{a \times b}$. Applying this rule, we get $(3^{1/4})^7 = 3^{(1/4) \times 7} = 3^{7/4}$. This step is crucial because it simplifies the expression into a single base raised to a single exponent, which is what we were aiming for.

3. Identify the equivalent expression:

   So, $(\sqrt[4]{3})^7$ simplifies to $3^{7/4}$. Looking at the given options:

   • A. $4^{3/7}$
   • B. $4^{7/3}$
   • C. $3^{7/4}$
   • D. $3^{4/7}$

   We can see that option C, $3^{7/4}$, matches our simplified expression. Therefore, the equivalent expression to $(\sqrt[4]{3})^7$ is $3^{7/4}$. Always double-check your work to ensure you haven't made any errors in the conversion or simplification process.

Why Other Options Are Incorrect

Understanding why the other options are incorrect is just as important as knowing the correct answer. This helps reinforce your understanding of the concepts and prevents you from making similar mistakes in the future. Let's break down why options A, B, and D are not equivalent to $(\sqrt[4]{3})^7$.

Option A: $4^{3/7}$

This option is incorrect because it changes both the base and the exponent. The original expression involves the base 3, not 4. Additionally, the exponent $3/7$ does not result from correctly applying the rules of exponents to the given expression. It seems like a random combination of the numbers involved, without any mathematical justification. Always remember to keep the base consistent unless there is a valid reason to change it, such as simplification or substitution.

Option B: $4^{7/3}$

Similar to option A, this option incorrectly changes the base from 3 to 4. The exponent $7/3$ is also incorrect. This exponent might arise from mistakenly inverting the fraction or misunderstanding the order of operations. The correct exponent should be derived from taking the 4th root and raising to the 7th power, which leads to $7/4$, not $7/3$. Double-checking the base and the correct application of exponent rules can help avoid this error.

Option D: $3^{4/7}$

This option keeps the correct base of 3 but has an incorrect exponent. The exponent $4/7$ is the inverse of the correct exponent, $7/4$. This mistake might come from confusing the numerator and denominator when converting between radical and exponential forms or misapplying the power of a power rule. To avoid this, always remember that $(\sqrt[n]{x})^m = x^{m/n}$, not $x^{n/m}$. A simple check can confirm that $3^{4/7}$ is not equivalent to $(\sqrt[4]{3})^7$.

Key Takeaways

Alright, guys, let's wrap things up with the key takeaways from this problem. Understanding these points will not only help you solve similar problems but also solidify your understanding of exponents and radicals.

• Converting Radicals to Exponents:

  The most important takeaway is the ability to convert radicals to fractional exponents. Remember that $\sqrt[n]{x} = x^{1/n}$. This conversion is essential for simplifying expressions involving radicals because it allows you to apply the rules of exponents. Practice converting different radicals to exponents to become comfortable with this process. For example, $\sqrt[5]{32} = 32^{1/5} = 2$.

• Power of a Power Rule:

  The power of a power rule, $(x^a)^b = x^{a \times b}$, is another critical concept. This rule is frequently used when simplifying expressions where an exponent is raised to another exponent. Make sure you understand how to apply this rule correctly. For instance, $(5^2)^3 = 5^{2 \times 3} = 5^6 = 15625$.

• Attention to Detail:

  Pay close attention to detail when simplifying expressions. A small mistake, such as inverting a fraction or misapplying a rule, can lead to an incorrect answer. Always double-check your work and ensure that each step is mathematically sound. This includes verifying the base, the exponents, and the correct application of rules.

• Understanding Incorrect Options:

  Understanding why the other options are incorrect is just as important as knowing the correct answer. This helps you avoid common mistakes and reinforces your understanding of the concepts. Analyze each incorrect option to identify the error and understand why it is not a valid solution. This practice can significantly improve your problem-solving skills.

Practice Problems

To further enhance your understanding, here are a few practice problems. Work through these problems on your own and then check your answers to reinforce what you’ve learned.

1. Simplify $(\sqrt[3]{5})^6$
2. Which expression is equivalent to $\sqrt[5]{x^{10}}$?
3. Rewrite $(\sqrt{7})^3$ using fractional exponents.

Conclusion

Simplifying expressions with radicals and exponents can seem challenging at first, but with a solid understanding of the basic principles and plenty of practice, you'll become a pro in no time! Remember to convert radicals to exponents, apply the power of a power rule, pay attention to detail, and understand why incorrect options are wrong. Keep practicing, and you'll master these concepts and excel in your math studies. You got this, guys!