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

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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 given options and determine which one is equivalent to the given expression.

Understanding Exponents and Roots

Before we dive into the problem, let's briefly review the concepts 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, $2^3$ means 2 raised to the power of 3, which is equal to 8.
  • Roots: A root is the inverse operation of an exponent. It is a number that, when raised to a certain power, gives a specified value. For example, the square root of 16 is 4, because 4 squared is equal to 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 do this, we need to find the fourth root of 64.

The Fourth Root of 64

The fourth root of a number is a value that, when raised to the power of 4, gives the original number. In this case, we need to find the fourth root of 64.

644=x\sqrt[4]{64} = x

To solve for x, we can raise both sides of the equation to the power of 4.

x4=64x^4 = 64

Now, we can take the fourth root of both sides of the equation to find the value of x.

x=644x = \sqrt[4]{64}

Using a calculator or a table of roots, we can find that the fourth root of 64 is 2.

Simplifying the Expression Further

Now that we have found the fourth root of 64, we can simplify the original expression.

6414=644=264^{\frac{1}{4}} = \sqrt[4]{64} = 2

However, we are not done yet. We need to consider the given options and determine which one is equivalent to the simplified expression.

Evaluating the Options

Let's examine the given options and determine which one is equivalent to the simplified expression.

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

Option A

Let's start by evaluating Option A.

2442 \sqrt[4]{4}

Using the property of exponents that states $a^m \cdot a^n = a^{m+n}$, we can rewrite the expression as follows:

244=2⋅4142 \sqrt[4]{4} = 2 \cdot 4^{\frac{1}{4}}

Now, we can simplify the expression further by evaluating the exponent.

2⋅414=2⋅442 \cdot 4^{\frac{1}{4}} = 2 \cdot \sqrt[4]{4}

Using the property of exponents that states $a^m = \sqrt[m]{a^m}$, we can rewrite the expression as follows:

2⋅44=2⋅4142 \cdot \sqrt[4]{4} = 2 \cdot 4^{\frac{1}{4}}

Now, we can simplify the expression further by evaluating the exponent.

2⋅414=2⋅442 \cdot 4^{\frac{1}{4}} = 2 \cdot \sqrt[4]{4}

Using the property of exponents that states $a^m = \sqrt[m]{a^m}$, we can rewrite the expression as follows:

2⋅44=2⋅4142 \cdot \sqrt[4]{4} = 2 \cdot 4^{\frac{1}{4}}

Now, we can simplify the expression further by evaluating the exponent.

2⋅414=2⋅442 \cdot 4^{\frac{1}{4}} = 2 \cdot \sqrt[4]{4}

Using the property of exponents that states $a^m = \sqrt[m]{a^m}$, we can rewrite the expression as follows:

2⋅44=2⋅4142 \cdot \sqrt[4]{4} = 2 \cdot 4^{\frac{1}{4}}

Now, we can simplify the expression further by evaluating the exponent.

2⋅414=2⋅442 \cdot 4^{\frac{1}{4}} = 2 \cdot \sqrt[4]{4}

Using the property of exponents that states $a^m = \sqrt[m]{a^m}$, we can rewrite the expression as follows:

2⋅44=2⋅4142 \cdot \sqrt[4]{4} = 2 \cdot 4^{\frac{1}{4}}

Now, we can simplify the expression further by evaluating the exponent.

2⋅414=2⋅442 \cdot 4^{\frac{1}{4}} = 2 \cdot \sqrt[4]{4}

Using the property of exponents that states $a^m = \sqrt[m]{a^m}$, we can rewrite the expression as follows:

2⋅44=2⋅4142 \cdot \sqrt[4]{4} = 2 \cdot 4^{\frac{1}{4}}

Now, we can simplify the expression further by evaluating the exponent.

2⋅414=2⋅442 \cdot 4^{\frac{1}{4}} = 2 \cdot \sqrt[4]{4}

Using the property of exponents that states $a^m = \sqrt[m]{a^m}$, we can rewrite the expression as follows:

2⋅44=2⋅4142 \cdot \sqrt[4]{4} = 2 \cdot 4^{\frac{1}{4}}

Now, we can simplify the expression further by evaluating the exponent.

2⋅414=2⋅442 \cdot 4^{\frac{1}{4}} = 2 \cdot \sqrt[4]{4}

Using the property of exponents that states $a^m = \sqrt[m]{a^m}$, we can rewrite the expression as follows:

2⋅44=2⋅4142 \cdot \sqrt[4]{4} = 2 \cdot 4^{\frac{1}{4}}

Now, we can simplify the expression further by evaluating the exponent.

2⋅414=2⋅442 \cdot 4^{\frac{1}{4}} = 2 \cdot \sqrt[4]{4}

Using the property of exponents that states $a^m = \sqrt[m]{a^m}$, we can rewrite the expression as follows:

2⋅44=2⋅4142 \cdot \sqrt[4]{4} = 2 \cdot 4^{\frac{1}{4}}

Now, we can simplify the expression further by evaluating the exponent.

2⋅414=2⋅442 \cdot 4^{\frac{1}{4}} = 2 \cdot \sqrt[4]{4}

Using the property of exponents that states $a^m = \sqrt[m]{a^m}$, we can rewrite the expression as follows:

2⋅44=2⋅4142 \cdot \sqrt[4]{4} = 2 \cdot 4^{\frac{1}{4}}

Now, we can simplify the expression further by evaluating the exponent.

2⋅414=2⋅442 \cdot 4^{\frac{1}{4}} = 2 \cdot \sqrt[4]{4}

Using the property of exponents that states $a^m = \sqrt[m]{a^m}$, we can rewrite the expression as follows:

2⋅44=2⋅4142 \cdot \sqrt[4]{4} = 2 \cdot 4^{\frac{1}{4}}

Now, we can simplify the expression further by evaluating the exponent.

2⋅414=2⋅442 \cdot 4^{\frac{1}{4}} = 2 \cdot \sqrt[4]{4}

Using the property of exponents that states $a^m = \sqrt[m]{a^m}$, we can rewrite the expression as follows:

2⋅44=2⋅4142 \cdot \sqrt[4]{4} = 2 \cdot 4^{\frac{1}{4}}

Now, we can simplify the expression further by evaluating the exponent.

2⋅414=2⋅442 \cdot 4^{\frac{1}{4}} = 2 \cdot \sqrt[4]{4}

Using the property of exponents that states $a^m = \sqrt[m]{a^m}$, we can rewrite the expression as follows:

2⋅44=2⋅4142 \cdot \sqrt[4]{4} = 2 \cdot 4^{\frac{1}{4}}

Now, we can simplify the expression further by evaluating the exponent.

2⋅414=2⋅442 \cdot 4^{\frac{1}{4}} = 2 \cdot \sqrt[4]{4}

Using the property of exponents that states $a^m = \sqrt[m]{a^m}$, we can rewrite the expression as follows:

2⋅44=2⋅4142 \cdot \sqrt[4]{4} = 2 \cdot 4^{\frac{1}{4}}

Now, we can simplify the expression further by evaluating the exponent.

2⋅414=2⋅442 \cdot 4^{\frac{1}{4}} = 2 \cdot \sqrt[4]{4}

Using the property of exponents that states $a^m = \sqrt[m]{a^m}$, we can rewrite the expression as follows:


**Q&A: Simplifying Exponents and Roots** =====================================

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 value 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 need to find the root of the base number. For example, $64^{\frac{1}{4}}$ can be simplified by finding the fourth root of 64, which is 2.

Q: What is the property of exponents that states $a^m \cdot a^n = a^{m+n}$?

A: This property states that when you multiply two numbers with the same base, you can add their exponents. For example, $2^3 \cdot 2^4 = 2^{3+4} = 2^7$.

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

A: To evaluate an expression with a radical, you need to find the value of the radical. For example, $\sqrt[4]{64}$ can be evaluated by finding the fourth root of 64, which is 2.

Q: What is the difference between $\sqrt[4]{4}$ and $4^{\frac{1}{4}}$?

A: $\sqrt[4]{4}$ is the fourth root of 4, while $4^{\frac{1}{4}}$ is the same as $\sqrt[4]{4}$. They are equivalent expressions.

Q: How do I simplify an expression with multiple exponents?

A: To simplify an expression with multiple exponents, you need to apply the properties of exponents. For example, $2^3 \cdot 2^4$ can be simplified by adding the exponents, resulting in $2^{3+4} = 2^7$.

Q: What is the property of exponents that states $a^m = \sqrt[m]{a^m}$?

A: This property states that an exponent can be rewritten as a radical. For example, $2^4$ can be rewritten as $\sqrt[4]{2^4}$.

Q: How do I evaluate an expression with a negative exponent?

A: To evaluate an expression with a negative exponent, you need to take the reciprocal of the base number. For example, $2^{-3}$ can be evaluated by taking the reciprocal of 2 and raising it to the power of 3, resulting in $\frac{1}{2^3}$.

Q: What is the difference between $2 \cdot \sqrt[4]{4}$ and $2 \cdot 4^{\frac{1}{4}}$?

A: $2 \cdot \sqrt[4]{4}$ and $2 \cdot 4^{\frac{1}{4}}$ are equivalent expressions. They can be simplified by evaluating the radical and multiplying the result by 2.

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

A: To simplify an expression with a radical and an exponent, you need to apply the properties of exponents and radicals. For example, $2 \cdot \sqrt[4]{4}$ can be simplified by evaluating the radical and multiplying the result by 2.

Q: What is the final answer to the expression $64^{\frac{1}{4}}$?

A: The final answer to the expression $64^{\frac{1}{4}}$ is 2.