Which Is Equivalent To $64^{\frac{1}{4}}$?A. $2 \sqrt[4]{4}$B. 4C. 16D. $15 \sqrt[4]{4}$

Introduction

Exponents and roots are fundamental concepts in mathematics that help us simplify complex expressions and solve equations. In this article, we will explore the concept of equivalent expressions, specifically focusing on the expression $64^{\frac{1}{4}}$. We will examine the options provided and determine which one is equivalent to the given expression.

Understanding Exponents and Roots

Before we dive into the problem, let's review the basics of exponents and roots.

• Exponents: An exponent is a small number that is written above and to the right of a number or a variable. It represents the power to which the base is raised. For example, in the expression $2^3$, the exponent 3 indicates that the base 2 is raised to the power of 3.
• Roots: A root is the inverse operation of an exponent. It represents the number that, when raised to a certain power, gives a specified value. For example, the fourth root of 16 is 2, because $2^4 = 16$.

Simplifying the Expression

Now that we have a basic understanding of exponents and roots, let's simplify the expression $64^{\frac{1}{4}}$.

To simplify this expression, we can use the property of exponents that states $a^{\frac{1}{n}} = \sqrt[n]{a}$. Applying this property to the given expression, we get:

$$64^{\frac{1}{4}} = \sqrt[4]{64}$$

Evaluating the Options

Now that we have simplified the expression, let's evaluate the options provided.

• Option A: $2 \sqrt[4]{4}$
• Option B: 4
• Option C: 16
• Option D: $15 \sqrt[4]{4}$

Option A

Let's start by evaluating Option A: $2 \sqrt[4]{4}$.

We can simplify this expression by evaluating the fourth root of 4:

$$\sqrt[4]{4} = 2^{\frac{1}{2}} = \sqrt{2}$$

Now, we can multiply this result by 2:

$$2 \sqrt[4]{4} = 2 \sqrt{2}$$

This expression is not equivalent to $64^{\frac{1}{4}}$, so we can eliminate Option A.

Option B

Next, let's evaluate Option B: 4.

We can simplify this expression by evaluating the fourth root of 64:

$$\sqrt[4]{64} = 2^{\frac{1}{2}} = \sqrt{2}$$

However, this result is not equal to 4, so we can eliminate Option B.

Option C

Now, let's evaluate Option C: 16.

We can simplify this expression by evaluating the fourth root of 64:

$$\sqrt[4]{64} = 2^{\frac{1}{2}} = \sqrt{2}$$

However, this result is not equal to 16, so we can eliminate Option C.

Option D

Finally, let's evaluate Option D: $15 \sqrt[4]{4}$.

We can simplify this expression by evaluating the fourth root of 4:

$$\sqrt[4]{4} = 2^{\frac{1}{2}} = \sqrt{2}$$

Now, we can multiply this result by 15:

$$15 \sqrt[4]{4} = 15 \sqrt{2}$$

This expression is not equivalent to $64^{\frac{1}{4}}$, so we can eliminate Option D.

Conclusion

After evaluating all the options, we can conclude that none of the options provided are equivalent to $64^{\frac{1}{4}}$. However, we can simplify the expression using the property of exponents that states $a^{\frac{1}{n}} = \sqrt[n]{a}$. Applying this property to the given expression, we get:

$$64^{\frac{1}{4}} = \sqrt[4]{64} = 2^{\frac{1}{2}} = \sqrt{2} \times \sqrt{2} = 2$$

Therefore, the correct answer is not among the options provided.

Final Answer

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Q: What is the difference between an exponent and a root?

A: An exponent is a small number that is written above and to the right of a number or a variable, representing the power to which the base is raised. A root, on the other hand, is the inverse operation of an exponent, representing the number that, when raised to a certain power, gives a specified value.

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

A: To simplify an expression with a fractional exponent, you can use the property of exponents that states $a^{\frac{1}{n}} = \sqrt[n]{a}$. This property allows you to rewrite the expression as a root, making it easier to simplify.

Q: What is the value of $64^{\frac{1}{4}}$?

A: To evaluate this expression, we can use the property of exponents that states $a^{\frac{1}{n}} = \sqrt[n]{a}$. Applying this property to the given expression, we get:

$$64^{\frac{1}{4}} = \sqrt[4]{64} = 2^{\frac{1}{2}} = \sqrt{2} \times \sqrt{2} = 2$$

Q: How do I evaluate an expression with a radical?

A: To evaluate an expression with a radical, you can start by simplifying the radical. If the radical is a perfect square, you can simplify it by taking the square root of the number inside the radical. For example:

$$\sqrt{16} = \sqrt{4 \times 4} = \sqrt{4} \times \sqrt{4} = 2 \times 2 = 4$$

Q: What is the difference between a rational exponent and an irrational exponent?

A: A rational exponent is an exponent that can be expressed as a fraction, such as $\frac{1}{2}$ or $\frac{3}{4}$. An irrational exponent, on the other hand, is an exponent that cannot be expressed as a fraction, such as $\sqrt{2}$ or $\pi$.

Q: How do I simplify an expression with a rational exponent?

A: To simplify an expression with a rational exponent, you can use the property of exponents that states $a^{\frac{m}{n}} = (a^m){\frac{1}{n}}$. This property allows you to rewrite the expression as a root, making it easier to simplify.

Q: What is the value of $2^{\frac{3}{4}}$?

A: To evaluate this expression, we can use the property of exponents that states $a^{\frac{m}{n}} = (a^m){\frac{1}{n}}$. Applying this property to the given expression, we get:

$$2^{\frac{3}{4}} = (2^3)^{\frac{1}{4}} = 8^{\frac{1}{4}} = \sqrt[4]{8} = \sqrt[4]{2^3} = 2^{\frac{3}{4}}$$

Conclusion

In this article, we have covered some common questions and answers related to simplifying exponents and roots. We have discussed the difference between an exponent and a root, how to simplify expressions with fractional exponents, and how to evaluate expressions with radicals. We have also covered the difference between rational and irrational exponents, and how to simplify expressions with rational exponents. By following these tips and techniques, you can simplify complex expressions and solve equations with ease.

Final Answer

The final answer is: $\boxed{2}$