The Expression $16^{-\frac{3}{4}}$ Is Equivalent To Which Of The Following Expressions?A. $(\sqrt[4]{16})^3$B. \$(\sqrt[4]{16})^{-3}$[/tex\]C. $(\sqrt[3]{16})^{-4}$D.

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Introduction

In mathematics, expressions involving exponents and roots are fundamental concepts that play a crucial role in various mathematical operations. The expression $16^{-\frac{3}{4}}$ is a classic example of an exponentiated expression that can be simplified using various mathematical techniques. In this article, we will delve into the world of exponents and roots to determine which of the given expressions is equivalent to $16^{-\frac{3}{4}}$.

Understanding Exponents and Roots

Before we proceed with the analysis, it is essential to understand the concepts of exponents and roots. An exponent is a small number that is raised to a power, indicating how many times the base is multiplied by itself. For example, $a^b$ means $a$ multiplied by itself $b$ times. On the other hand, a root is the inverse operation of an exponent, which involves finding the number that, when raised to a certain power, gives a specified value.

Simplifying the Expression $16^{-\frac{3}{4}}$

To simplify the expression $16^{-\frac{3}{4}}$, we can start by rewriting it as a fraction:

$$16^{-\frac{3}{4}} = \frac{1}{16^{\frac{3}{4}}}$$

Now, we can simplify the denominator using the rule of exponents that states $a^m \cdot a^n = a^{m+n}$. In this case, we have:

$$16^{\frac{3}{4}} = (2^4)^{\frac{3}{4}} = 2^{4 \cdot \frac{3}{4}} = 2^3 = 8$$

Therefore, the expression $16^{-\frac{3}{4}}$ can be simplified as:

$$16^{-\frac{3}{4}} = \frac{1}{8}$$

Comparing with the Given Options

Now that we have simplified the expression $16^{-\frac{3}{4}}$, we can compare it with the given options to determine which one is equivalent.

Option A: $(\sqrt[4]{16})^3$

To simplify this expression, we can start by evaluating the fourth root of 16:

$$\sqrt[4]{16} = 2$$

Now, we can raise this value to the power of 3:

$$(\sqrt[4]{16})^3 = 2^3 = 8$$

This is not equivalent to the simplified expression $16^{-\frac{3}{4}}$, which is $\frac{1}{8}$.

Option B: $(\sqrt[4]{16})^{-3}$

To simplify this expression, we can start by evaluating the fourth root of 16:

$$\sqrt[4]{16} = 2$$

Now, we can raise this value to the power of -3:

$$(\sqrt[4]{16})^{-3} = 2^{-3} = \frac{1}{2^3} = \frac{1}{8}$$

This is equivalent to the simplified expression $16^{-\frac{3}{4}}$, which is $\frac{1}{8}$.

Option C: $(\sqrt[3]{16})^{-4}$

To simplify this expression, we can start by evaluating the cube root of 16:

$$\sqrt[3]{16} = 2$$

Now, we can raise this value to the power of -4:

$$(\sqrt[3]{16})^{-4} = 2^{-4} = \frac{1}{2^4} = \frac{1}{16}$$

This is not equivalent to the simplified expression $16^{-\frac{3}{4}}$, which is $\frac{1}{8}$.

Option D: $(\sqrt[4]{16})^4$

To simplify this expression, we can start by evaluating the fourth root of 16:

$$\sqrt[4]{16} = 2$$

Now, we can raise this value to the power of 4:

$$(\sqrt[4]{16})^4 = 2^4 = 16$$

This is not equivalent to the simplified expression $16^{-\frac{3}{4}}$, which is $\frac{1}{8}$.

Conclusion

In conclusion, the expression $16^{-\frac{3}{4}}$ is equivalent to $(\sqrt[4]{16})^{-3}$. This is because both expressions simplify to $\frac{1}{8}$. The other options, $(\sqrt[4]{16})^3$, $(\sqrt[3]{16})^{-4}$, and $(\sqrt[4]{16})^4$, are not equivalent to the simplified expression $16^{-\frac{3}{4}}$.

Final Answer

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Frequently Asked Questions

In the previous article, we analyzed the expression $16^{-\frac{3}{4}}$ and determined that it is equivalent to $(\sqrt[4]{16})^{-3}$. However, we understand that some readers may still have questions about this topic. In this article, we will address some of the most frequently asked questions about the expression $16^{-\frac{3}{4}}$.

Q: What is the meaning of the exponent $-\frac{3}{4}$?

A: The exponent $-\frac{3}{4}$ indicates that the base, 16, is being raised to the power of $-\frac{3}{4}$. This means that the base is being multiplied by itself $-\frac{3}{4}$ times.

Q: How do I simplify the expression $16^{-\frac{3}{4}}$?

A: To simplify the expression $16^{-\frac{3}{4}}$, you can start by rewriting it as a fraction:

$$16^{-\frac{3}{4}} = \frac{1}{16^{\frac{3}{4}}}$$

Then, you can simplify the denominator using the rule of exponents that states $a^m \cdot a^n = a^{m+n}$. In this case, we have:

$$16^{\frac{3}{4}} = (2^4)^{\frac{3}{4}} = 2^{4 \cdot \frac{3}{4}} = 2^3 = 8$$

Therefore, the expression $16^{-\frac{3}{4}}$ can be simplified as:

$$16^{-\frac{3}{4}} = \frac{1}{8}$$

Q: What is the relationship between the expression $16^{-\frac{3}{4}}$ and the expression $(\sqrt[4]{16})^{-3}$?

A: The expression $16^{-\frac{3}{4}}$ is equivalent to the expression $(\sqrt[4]{16})^{-3}$. This is because both expressions simplify to $\frac{1}{8}$.

Q: How do I evaluate the expression $(\sqrt[4]{16})^{-3}$?

A: To evaluate the expression $(\sqrt[4]{16})^{-3}$, you can start by evaluating the fourth root of 16:

$$\sqrt[4]{16} = 2$$

Then, you can raise this value to the power of -3:

$$(\sqrt[4]{16})^{-3} = 2^{-3} = \frac{1}{2^3} = \frac{1}{8}$$

Q: What is the significance of the expression $16^{-\frac{3}{4}}$?

A: The expression $16^{-\frac{3}{4}}$ is a classic example of an exponentiated expression that can be simplified using various mathematical techniques. It is an important concept in mathematics that can be used to solve a wide range of problems.

Q: How can I apply the concept of the expression $16^{-\frac{3}{4}}$ to real-world problems?

A: The concept of the expression $16^{-\frac{3}{4}}$ can be applied to a wide range of real-world problems, such as:

• Calculating the area of a circle
• Determining the volume of a sphere
• Finding the surface area of a cube
• Solving problems involving exponents and roots

Conclusion

In conclusion, the expression $16^{-\frac{3}{4}}$ is a fundamental concept in mathematics that can be simplified using various mathematical techniques. We hope that this Q&A article has provided you with a better understanding of this concept and how it can be applied to real-world problems.

Final Answer

The final answer is $\boxed{B}$