Finding the Largest Perfect Fourth Power

Oct 13, 2025 by thesweetdesigns.com 76 views

Introduction

Radicals are a fundamental concept in mathematics, and simplifying them is an essential skill for any math enthusiast. In this article, we will explore the simplified radical form of $\sqrt[4]{48}$ , which is a fourth root of a number. We will break down the process step by step, making it easy to understand and follow.

Understanding the Problem

The problem asks us to simplify the fourth root of 48, denoted as $\sqrt[4]{48}$ . To simplify this expression, we need to find the largest perfect fourth power that divides 48. This will help us to rewrite the expression in a simpler form.

Breaking Down the Number

Let's start by breaking down the number 48 into its prime factors. We can write 48 as:

48 = 2 × 2 × 2 × 2 × 3

Finding the Largest Perfect Fourth Power

Now, let's find the largest perfect fourth power that divides 48. A perfect fourth power is a number that can be expressed as the fourth power of an integer. In this case, we can see that 16 is a perfect fourth power, since 16 = 2 × 2 × 2 × 2.

Simplifying the Expression

Now that we have found the largest perfect fourth power that divides 48, we can simplify the expression. We can rewrite $\sqrt[4]{48}$ as:

$\sqrt[4]{48}$ = $\sqrt[4]{2^4 × 3}$

Applying the Property of Radicals

We can use the property of radicals that states $\sqrt[n]{a^n} = a$ . In this case, we can simplify the expression as:

$\sqrt[4]{2^4 × 3}$ = 2 × $\sqrt[4]{3}$

Simplifying the Radical

Now, let's simplify the radical $\sqrt[4]{3}$ . We can rewrite it as:

$\sqrt[4]{3}$ = $\sqrt[4]{3^1}$

Applying the Property of Radicals Again

We can use the property of radicals again to simplify the expression:

$\sqrt[4]{3^1}$ = $\sqrt[4]{3}$

Final Answer

Therefore, the simplified radical form of $\sqrt[4]{48}$ is:

$\sqrt[4]{48}$ = 2 × $\sqrt[4]{3}$

Conclusion

In this article, we have simplified the fourth root of 48, denoted as $\sqrt[4]{48}$ . We have broken down the number 48 into its prime factors, found the largest perfect fourth power that divides 48, and applied the property of radicals to simplify the expression. The final answer is 2 × $\sqrt[4]{3}$ .

Frequently Asked Questions

What is the simplified radical form of $\sqrt[4]{48}$ ?
How do we simplify the fourth root of a number?
What is the largest perfect fourth power that divides 48?

Answer to FAQs

The simplified radical form of $\sqrt[4]{48}$ is 2 × $\sqrt[4]{3}$ .
To simplify the fourth root of a number, we need to find the largest perfect fourth power that divides the number.
The largest perfect fourth power that divides 48 is 16.

Additional Resources

For more information on radicals, please visit the following website: [insert website URL]
For more practice problems on simplifying radicals, please visit the following website: [insert website URL]

Final Thoughts

Simplifying radicals is an essential skill for any math enthusiast. By following the steps outlined in this article, you can simplify the fourth root of any number. Remember to break down the number into its prime factors, find the largest perfect fourth power that divides the number, and apply the property of radicals to simplify the expression. With practice and patience, you can become proficient in simplifying radicals and tackle even the most challenging math problems.

Introduction

In our previous article, we explored the simplified radical form of $\sqrt[4]{48}$ . We broke down the number 48 into its prime factors, found the largest perfect fourth power that divides 48, and applied the property of radicals to simplify the expression. In this article, we will answer some frequently asked questions related to the simplified radical form of $\sqrt[4]{48}$ .

Q&A

Q: What is the simplified radical form of $\sqrt[4]{48}$ ?

A: The simplified radical form of $\sqrt[4]{48}$ is 2 × $\sqrt[4]{3}$ .

Q: How do we simplify the fourth root of a number?

A: To simplify the fourth root of a number, we need to find the largest perfect fourth power that divides the number. We can then rewrite the expression using the property of radicals.

Q: What is the largest perfect fourth power that divides 48?

A: The largest perfect fourth power that divides 48 is 16.

Q: Can we simplify the radical $\sqrt[4]{3}$ further?

A: No, we cannot simplify the radical $\sqrt[4]{3}$ further. It is already in its simplest form.

Q: How do we know that 2 is the correct coefficient?

A: We know that 2 is the correct coefficient because it is the fourth root of 16, which is the largest perfect fourth power that divides 48.

Q: Can we use this method to simplify other radicals?

A: Yes, we can use this method to simplify other radicals. We need to find the largest perfect power that divides the number, and then apply the property of radicals to simplify the expression.

Q: What are some common mistakes to avoid when simplifying radicals?

A: Some common mistakes to avoid when simplifying radicals include:

Not finding the largest perfect power that divides the number
Not applying the property of radicals correctly
Not simplifying the radical further if possible

Q: How can we check our answer?

A: We can check our answer by raising the simplified expression to the power of 4 and seeing if it equals the original number.

Conclusion

In this article, we have answered some frequently asked questions related to the simplified radical form of $\sqrt[4]{48}$ . We have also discussed some common mistakes to avoid when simplifying radicals and how to check our answer. By following the steps outlined in this article, you can simplify the fourth root of any number and become proficient in simplifying radicals.

Additional Resources

For more information on radicals, please visit the following website: [insert website URL]
For more practice problems on simplifying radicals, please visit the following website: [insert website URL]

Final Thoughts

Simplifying radicals is an essential skill for any math enthusiast. By following the steps outlined in this article, you can simplify the fourth root of any number and become proficient in simplifying radicals. Remember to find the largest perfect power that divides the number, apply the property of radicals correctly, and simplify the radical further if possible. With practice and patience, you can become proficient in simplifying radicals and tackle even the most challenging math problems.

Frequently Asked Questions

What is the simplified radical form of $\sqrt[4]{48}$ ?
How do we simplify the fourth root of a number?
What is the largest perfect fourth power that divides 48?
Can we simplify the radical $\sqrt[4]{3}$ further?
How do we know that 2 is the correct coefficient?
Can we use this method to simplify other radicals?
What are some common mistakes to avoid when simplifying radicals?
How can we check our answer?

Answer to FAQs

The simplified radical form of $\sqrt[4]{48}$ is 2 × $\sqrt[4]{3}$ .
To simplify the fourth root of a number, we need to find the largest perfect fourth power that divides the number.
The largest perfect fourth power that divides 48 is 16.
No, we cannot simplify the radical $\sqrt[4]{3}$ further.
We know that 2 is the correct coefficient because it is the fourth root of 16, which is the largest perfect fourth power that divides 48.
Yes, we can use this method to simplify other radicals.
Some common mistakes to avoid when simplifying radicals include not finding the largest perfect power that divides the number, not applying the property of radicals correctly, and not simplifying the radical further if possible.
We can check our answer by raising the simplified expression to the power of 4 and seeing if it equals the original number.