Which Expression Is Equivalent To ( 5 3 4 ) 1 2 \left(\sqrt[4]{5^3}\right)^{\frac{1}{2}} ( 4 5 3 ​ ) 2 1 ​ ?A. 5 6 5^6 5 6 B. 5 2 3 5^{\frac{2}{3}} 5 3 2 ​ C. 5 16 3 5^{\frac{16}{3}} 5 3 16 ​ D. 5 3 8 5^{\frac{3}{8}} 5 8 3 ​

Introduction

Radical expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill for students to master. In this article, we will explore the process of simplifying radical expressions, with a focus on the given expression $\left(\sqrt[4]{5^3}\right)^{\frac{1}{2}}$. We will break down the expression into manageable parts, apply the rules of exponents, and arrive at the equivalent expression.

Understanding the Given Expression

The given expression is $\left(\sqrt[4]{5^3}\right)^{\frac{1}{2}}$. To simplify this expression, we need to understand the properties of radicals and exponents. The expression involves a fourth root and a square root, which can be simplified using the rules of exponents.

Breaking Down the Expression

Let's break down the expression into smaller parts:

• $\sqrt[4]{5^3}$: This is a fourth root of $5^3$. To simplify this expression, we need to apply the rule of exponents, which states that $\sqrt[n]{a^n} = a$.
• $\left(\sqrt[4]{5^3}\right)^{\frac{1}{2}}$: This is a square root of the fourth root of $5^3$. To simplify this expression, we need to apply the rule of exponents, which states that $\left(a^m\right)^n = a^{mn}$.

Applying the Rules of Exponents

Now, let's apply the rules of exponents to simplify the expression:

• $\sqrt[4]{5^3} = 5^{\frac{3}{4}}$: This is because the fourth root of $5^3$ is equal to $5$ raised to the power of $\frac{3}{4}$.
• $\left(5^{\frac{3}{4}}\right)^{\frac{1}{2}} = 5^{\frac{3}{8}}$: This is because the square root of $5^{\frac{3}{4}}$ is equal to $5$ raised to the power of $\frac{3}{8}$.

Conclusion

In conclusion, the expression $\left(\sqrt[4]{5^3}\right)^{\frac{1}{2}}$ is equivalent to $5^{\frac{3}{8}}$. This is because we applied the rules of exponents to simplify the expression, and arrived at the equivalent expression.

Answer

The correct answer is D. $5^{\frac{3}{8}}$.

Discussion

Radical expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill for students to master. In this article, we explored the process of simplifying radical expressions, with a focus on the given expression $\left(\sqrt[4]{5^3}\right)^{\frac{1}{2}}$. We broke down the expression into manageable parts, applied the rules of exponents, and arrived at the equivalent expression.

Tips and Tricks

Here are some tips and tricks to help you simplify radical expressions:

• Always start by simplifying the expression inside the radical sign.
• Apply the rules of exponents to simplify the expression.
• Use the property of radicals to simplify the expression.
• Check your work by plugging in values to ensure that the expression is equivalent to the original expression.

Common Mistakes

Here are some common mistakes to avoid when simplifying radical expressions:

• Not simplifying the expression inside the radical sign.
• Not applying the rules of exponents.
• Not using the property of radicals to simplify the expression.
• Not checking your work by plugging in values.

Real-World Applications

Radical expressions have many real-world applications, including:

• Physics: Radical expressions are used to describe the motion of objects in physics.
• Engineering: Radical expressions are used to describe the behavior of electrical circuits in engineering.
• Computer Science: Radical expressions are used to describe the behavior of algorithms in computer science.

Conclusion

Introduction

Radical expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill for students to master. In our previous article, we explored the process of simplifying radical expressions, with a focus on the given expression $\left(\sqrt[4]{5^3}\right)^{\frac{1}{2}}$. In this article, we will provide a Q&A guide to help you understand and simplify radical expressions.

Q: What is a radical expression?

A: A radical expression is an expression that involves a root, such as a square root, cube root, or fourth root.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to apply the rules of exponents and use the property of radicals. Start by simplifying the expression inside the radical sign, then apply the rules of exponents to simplify the expression.

Q: What is the rule of exponents for radicals?

A: The rule of exponents for radicals states that $\sqrt[n]{a^n} = a$.

Q: How do I apply the rule of exponents for radicals?

A: To apply the rule of exponents for radicals, you need to simplify the expression inside the radical sign by raising it to the power of $\frac{1}{n}$, where $n$ is the index of the radical.

Q: What is the property of radicals?

A: The property of radicals states that $\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{ab}$.

Q: How do I use the property of radicals to simplify an expression?

A: To use the property of radicals to simplify an expression, you need to multiply the expressions inside the radical signs and then simplify the resulting expression.

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

A: Some common mistakes to avoid when simplifying radical expressions include:

• Not simplifying the expression inside the radical sign.
• Not applying the rules of exponents.
• Not using the property of radicals to simplify the expression.
• Not checking your work by plugging in values.

Q: How do I check my work when simplifying radical expressions?

A: To check your work when simplifying radical expressions, you need to plug in values to ensure that the expression is equivalent to the original expression.

Q: What are some real-world applications of radical expressions?

A: Radical expressions have many real-world applications, including:

• Physics: Radical expressions are used to describe the motion of objects in physics.
• Engineering: Radical expressions are used to describe the behavior of electrical circuits in engineering.
• Computer Science: Radical expressions are used to describe the behavior of algorithms in computer science.

Q: How do I simplify a radical expression with a negative exponent?

A: To simplify a radical expression with a negative exponent, you need to apply the rule of exponents, which states that $a^{-n} = \frac{1}{a^n}$.

Q: How do I simplify a radical expression with a fractional exponent?

A: To simplify a radical expression with a fractional exponent, you need to apply the rule of exponents, which states that $a^{\frac{m}{n}} = \sqrt[n]{a^m}$.

Conclusion

In conclusion, simplifying radical expressions is a crucial skill for students to master. By applying the rules of exponents and using the property of radicals, we can simplify radical expressions and arrive at the equivalent expression. Remember to always start by simplifying the expression inside the radical sign, apply the rules of exponents, and use the property of radicals to simplify the expression.