{ \left(\frac{27}{64}\right)^{-\frac{2}{3}}=?$}$A. { -\frac{9}{16}$}$B. { -\frac{9}{32}$}$C. { \frac{9}{32}$}$D. { \frac{16}{9}$}$E. { \frac{32}{9}$}$

Introduction

In this article, we will delve into the world of exponents and explore a specific problem that involves a negative exponent. The problem is as follows: $\left(\frac{27}{64}\right)^{-\frac{2}{3}}=?$. We will break down the solution step by step, using mathematical concepts and formulas to arrive at the correct answer.

Understanding Negative Exponents

Before we dive into the problem, let's take a moment to understand what negative exponents mean. A negative exponent is a shorthand way of writing a fraction with a negative power. For example, $a^{-n} = \frac{1}{a^n}$. This means that when we see a negative exponent, we can rewrite it as a fraction with a positive exponent in the denominator.

Breaking Down the Problem

Now that we have a basic understanding of negative exponents, let's break down the problem at hand. We are given the expression $\left(\frac{27}{64}\right)^{-\frac{2}{3}}$. To solve this problem, we need to apply the rule for negative exponents, which states that $a^{-n} = \frac{1}{a^n}$.

Step 1: Apply the Rule for Negative Exponents

Using the rule for negative exponents, we can rewrite the expression as follows:

$$\left(\frac{27}{64}\right)^{-\frac{2}{3}} = \frac{1}{\left(\frac{27}{64}\right)^{\frac{2}{3}}}$$

Step 2: Simplify the Expression

Now that we have rewritten the expression, let's simplify it further. To do this, we need to apply the rule for fractional exponents, which states that $a^{\frac{m}{n}} = \sqrt[n]{a^m}$. In this case, we have $\left(\frac{27}{64}\right)^{\frac{2}{3}}$, which can be rewritten as $\sqrt[3]{\left(\frac{27}{64}\right)^2}$.

Step 3: Evaluate the Expression

Now that we have simplified the expression, let's evaluate it. To do this, we need to calculate the value of $\sqrt[3]{\left(\frac{27}{64}\right)^2}$. We can do this by first calculating the value of $\left(\frac{27}{64}\right)^2$, and then taking the cube root of the result.

Step 4: Calculate the Value of $\left(\frac{27}{64}\right)^2$

To calculate the value of $\left(\frac{27}{64}\right)^2$, we can simply multiply the numerator and denominator by themselves:

$$\left(\frac{27}{64}\right)^2 = \frac{27 \cdot 27}{64 \cdot 64} = \frac{729}{4096}$$

Step 5: Take the Cube Root of the Result

Now that we have calculated the value of $\left(\frac{27}{64}\right)^2$, let's take the cube root of the result. To do this, we can use the fact that $\sqrt[3]{a^3} = a$. In this case, we have $\sqrt[3]{\frac{729}{4096}}$, which can be rewritten as $\frac{\sqrt[3]{729}}{\sqrt[3]{4096}}$.

Step 6: Simplify the Expression

Now that we have taken the cube root of the result, let's simplify the expression further. To do this, we need to calculate the value of $\sqrt[3]{729}$ and $\sqrt[3]{4096}$.

Step 7: Calculate the Value of $\sqrt[3]{729}$

To calculate the value of $\sqrt[3]{729}$, we can use the fact that $729 = 9^3$. Therefore, we can rewrite $\sqrt[3]{729}$ as $\sqrt[3]{9^3}$, which can be simplified to $9$.

Step 8: Calculate the Value of $\sqrt[3]{4096}$

To calculate the value of $\sqrt[3]{4096}$, we can use the fact that $4096 = 16^3$. Therefore, we can rewrite $\sqrt[3]{4096}$ as $\sqrt[3]{16^3}$, which can be simplified to $16$.

Step 9: Simplify the Expression

Now that we have calculated the value of $\sqrt[3]{729}$ and $\sqrt[3]{4096}$, let's simplify the expression further. To do this, we can substitute the values we calculated into the expression:

$$\frac{\sqrt[3]{729}}{\sqrt[3]{4096}} = \frac{9}{16}$$

Conclusion

In this article, we solved the exponent problem $\left(\frac{27}{64}\right)^{-\frac{2}{3}}$ using mathematical concepts and formulas. We broke down the problem step by step, applying the rule for negative exponents and simplifying the expression using fractional exponents. We then evaluated the expression, calculating the value of $\sqrt[3]{\left(\frac{27}{64}\right)^2}$ and taking the cube root of the result. Finally, we simplified the expression, arriving at the correct answer of $\frac{9}{16}$.

Answer

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Q: What is an exponent?

A: An exponent is a small number that is written to the right and above a base number. It indicates how many times the base number should be multiplied by itself. For example, in the expression $a^b$, $a$ is the base and $b$ is the exponent.

Q: What is a negative exponent?

A: A negative exponent is a shorthand way of writing a fraction with a negative power. For example, $a^{-n} = \frac{1}{a^n}$. This means that when we see a negative exponent, we can rewrite it as a fraction with a positive exponent in the denominator.

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

A: To simplify an expression with a negative exponent, we can apply the rule for negative exponents, which states that $a^{-n} = \frac{1}{a^n}$. We can then simplify the resulting fraction by dividing the numerator and denominator by their greatest common divisor.

Q: What is the difference between a negative exponent and a fraction with a negative power?

A: A negative exponent is a shorthand way of writing a fraction with a negative power. For example, $a^{-n} = \frac{1}{a^n}$. A fraction with a negative power, on the other hand, is a fraction where the power is negative. For example, $\frac{1}{a^n}$ is a fraction with a negative power, but it is not the same as a negative exponent.

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

A: To evaluate an expression with a negative exponent, we can apply the rule for negative exponents, which states that $a^{-n} = \frac{1}{a^n}$. We can then simplify the resulting fraction by dividing the numerator and denominator by their greatest common divisor.

Q: What is the relationship between exponents and roots?

A: Exponents and roots are related in that they are inverse operations. For example, $a^b$ and $\sqrt[b]{a}$ are inverse operations. This means that if we raise a number to a power, we can take the root of the result to get back to the original number.

Q: How do I simplify an expression with a cube root?

A: To simplify an expression with a cube root, we can use the fact that $\sqrt[3]{a^3} = a$. We can then simplify the resulting expression by multiplying the base number by itself three times.

Q: What is the difference between a cube root and a square root?

A: A cube root is a root that is raised to the power of 3, while a square root is a root that is raised to the power of 2. For example, $\sqrt[3]{a}$ is a cube root, while $\sqrt{a}$ is a square root.

Q: How do I evaluate an expression with a cube root?

A: To evaluate an expression with a cube root, we can use the fact that $\sqrt[3]{a^3} = a$. We can then simplify the resulting expression by multiplying the base number by itself three times.

Conclusion

In this article, we answered some of the most frequently asked questions about exponents and negative exponents. We covered topics such as the definition of an exponent, the rule for negative exponents, and the relationship between exponents and roots. We also provided examples and explanations to help illustrate the concepts. We hope that this article has been helpful in clarifying any confusion you may have had about exponents and negative exponents.