Evaluate The Expression: ( 27 64 ) − 2 3 = ? \left(\frac{27}{64}\right)^{-\frac{2}{3}}=? ( 64 27 ​ ) − 3 2 ​ = ? A. − 9 16 -\frac{9}{16} − 16 9 ​ B. − 9 32 -\frac{9}{32} − 32 9 ​ C. 9 32 \frac{9}{32} 32 9 ​ D. 16 10 \frac{16}{10} 10 16 ​ E. 32 9 \frac{32}{9} 9 32 ​

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Understanding the Problem

When dealing with exponents, it's essential to understand the rules and properties that govern them. In this problem, we're given the expression $\left(\frac{27}{64}\right)^{-\frac{2}{3}}$ and asked to evaluate it. To do this, we need to apply the rules of exponents, specifically the rule for negative exponents.

The Rule for Negative Exponents

A negative exponent is defined as the reciprocal of the positive exponent. In other words, $a^{-n} = \frac{1}{a^n}$. This rule can be applied to any expression with a negative exponent.

Applying the Rule to the Given Expression

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

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

Simplifying the Expression

To simplify the expression, we need to evaluate the exponent $\frac{2}{3}$. This means we need to raise both the numerator and the denominator to the power of $\frac{2}{3}$.

$$\left(\frac{27}{64}\right)^{\frac{2}{3}} = \left(\frac{3^3}{4^3}\right)^{\frac{2}{3}} = \frac{3^2}{4^2} = \frac{9}{16}$$

Substituting the Simplified Expression

Now that we have simplified the expression, we can substitute it back into the original expression:

$$\frac{1}{\left(\frac{27}{64}\right)^{\frac{2}{3}}} = \frac{1}{\frac{9}{16}} = \frac{16}{9}$$

Evaluating the Final Expression

However, we need to re-evaluate the expression $\left(\frac{27}{64}\right)^{-\frac{2}{3}}$.

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

$$\left(\frac{64}{27}\right)^{\frac{2}{3}} = \left(\frac{4^3}{3^3}\right)^{\frac{2}{3}} = \frac{4^2}{3^2} = \frac{16}{9}$$

$$\left(\frac{27}{64}\right)^{-\frac{2}{3}} = \frac{1}{\frac{16}{9}} = \frac{9}{16}$$

Conclusion

In conclusion, the correct answer to the expression $\left(\frac{27}{64}\right)^{-\frac{2}{3}}$ is $\frac{9}{16}$. This is the reciprocal of the simplified expression, which is $\frac{16}{9}$.

Final Answer

The final answer is $\boxed{\frac{9}{16}}$.

Discussion

This problem requires a good understanding of the rules of exponents, specifically the rule for negative exponents. It's essential to apply the rule correctly and simplify the expression step by step. The final answer is $\frac{9}{16}$.

Related Problems

• Evaluate the expression: $\left(\frac{8}{27}\right)^{-\frac{3}{2}}$
• Simplify the expression: $\left(\frac{2}{3}\right)^{-4}$
• Evaluate the expression: $\left(\frac{9}{16}\right)^{-\frac{1}{2}}$

Common Mistakes

• Forgetting to apply the rule for negative exponents
• Not simplifying the expression correctly
• Not evaluating the final expression correctly

Tips and Tricks

• Make sure to apply the rule for negative exponents correctly
• Simplify the expression step by step
• Evaluate the final expression carefully

Conclusion

In conclusion, the correct answer to the expression $\left(\frac{27}{64}\right)^{-\frac{2}{3}}$ is $\frac{9}{16}$. This requires a good understanding of the rules of exponents, specifically the rule for negative exponents. It's essential to apply the rule correctly and simplify the expression step by step.

Q: What is a negative exponent?

A: A negative exponent is defined as the reciprocal of the positive exponent. In other words, $a^{-n} = \frac{1}{a^n}$.

Q: How do I apply the rule for negative exponents?

A: To apply the rule for negative exponents, you need to rewrite the expression with a positive exponent. For example, $\left(\frac{27}{64}\right)^{-\frac{2}{3}} = \frac{1}{\left(\frac{27}{64}\right)^{\frac{2}{3}}}$.

Q: What is the difference between a negative exponent and a positive exponent?

A: A negative exponent is the reciprocal of the positive exponent. For example, $\left(\frac{27}{64}\right)^{-\frac{2}{3}} = \frac{1}{\left(\frac{27}{64}\right)^{\frac{2}{3}}}$.

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

A: To simplify an expression with a negative exponent, you need to apply the rule for negative exponents and simplify the expression step by step. For example, $\left(\frac{27}{64}\right)^{-\frac{2}{3}} = \frac{1}{\left(\frac{27}{64}\right)^{\frac{2}{3}}} = \frac{1}{\frac{9}{16}} = \frac{16}{9}$.

Q: What is the final answer to the expression $\left(\frac{27}{64}\right)^{-\frac{2}{3}}$?

A: The final answer to the expression $\left(\frac{27}{64}\right)^{-\frac{2}{3}}$ is $\frac{9}{16}$.

Q: What are some common mistakes to avoid when evaluating expressions with negative exponents?

A: Some common mistakes to avoid when evaluating expressions with negative exponents include:

• Forgetting to apply the rule for negative exponents
• Not simplifying the expression correctly
• Not evaluating the final expression correctly

Q: What are some tips and tricks for evaluating expressions with negative exponents?

A: Some tips and tricks for evaluating expressions with negative exponents include:

• Make sure to apply the rule for negative exponents correctly
• Simplify the expression step by step
• Evaluate the final expression carefully

Q: How do I evaluate the expression $\left(\frac{8}{27}\right)^{-\frac{3}{2}}$?

A: To evaluate the expression $\left(\frac{8}{27}\right)^{-\frac{3}{2}}$, you need to apply the rule for negative exponents and simplify the expression step by step. For example, $\left(\frac{8}{27}\right)^{-\frac{3}{2}} = \frac{1}{\left(\frac{8}{27}\right)^{\frac{3}{2}}} = \frac{1}{\frac{8^{\frac{3}{2}}}{27^{\frac{3}{2}}}} = \frac{27^{\frac{3}{2}}}{8^{\frac{3}{2}}} = \frac{27\sqrt{27}}{8\sqrt{8}} = \frac{3^3\sqrt{3^3}}{2^3\sqrt{2^3}} = \frac{3^3\cdot3^{\frac{3}{2}}}{2^3\cdot2^{\frac{3}{2}}} = \frac{3^{\frac{9}{2}}}{2^{\frac{9}{2}}} = \left(\frac{3}{2}\right)^{\frac{9}{2}}$

Q: How do I evaluate the expression $\left(\frac{9}{16}\right)^{-\frac{1}{2}}$?

A: To evaluate the expression $\left(\frac{9}{16}\right)^{-\frac{1}{2}}$, you need to apply the rule for negative exponents and simplify the expression step by step. For example, $\left(\frac{9}{16}\right)^{-\frac{1}{2}} = \frac{1}{\left(\frac{9}{16}\right)^{\frac{1}{2}}} = \frac{1}{\frac{3}{4}} = \frac{4}{3}$.

Q: What are some related problems to the expression $\left(\frac{27}{64}\right)^{-\frac{2}{3}}$?

A: Some related problems to the expression $\left(\frac{27}{64}\right)^{-\frac{2}{3}}$ include:

• Evaluate the expression: $\left(\frac{8}{27}\right)^{-\frac{3}{2}}$
• Simplify the expression: $\left(\frac{2}{3}\right)^{-4}$
• Evaluate the expression: $\left(\frac{9}{16}\right)^{-\frac{1}{2}}$

Q: What are some common mistakes to avoid when evaluating expressions with negative exponents?

A: Some common mistakes to avoid when evaluating expressions with negative exponents include:

• Forgetting to apply the rule for negative exponents
• Not simplifying the expression correctly
• Not evaluating the final expression correctly

Q: What are some tips and tricks for evaluating expressions with negative exponents?

A: Some tips and tricks for evaluating expressions with negative exponents include:

• Make sure to apply the rule for negative exponents correctly
• Simplify the expression step by step
• Evaluate the final expression carefully