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 9 \frac{16}{9} 9 16 ​ E. 32 9 \frac{32}{9} 9 32 ​

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Understanding the Problem

When evaluating an expression with a negative exponent, we need to apply the rule that states $a^{-n} = \frac{1}{a^n}$. This rule allows us to rewrite the expression with a positive exponent, making it easier to simplify.

Applying the Negative Exponent Rule

To evaluate the expression $\left(\frac{27}{64}\right)^{-\frac{2}{3}}$, we can apply the negative exponent rule by rewriting it as $\frac{1}{\left(\frac{27}{64}\right)^{\frac{2}{3}}}$.

Simplifying the Expression

Now, we need to simplify the expression inside the parentheses. To do this, we can use the rule that states $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$. Applying this rule, we get:

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

Evaluating the Exponents

Next, we need to evaluate the exponents $27^{\frac{2}{3}}$ and $64^{\frac{2}{3}}$. To do this, we can use the rule that states $a^{\frac{m}{n}} = \sqrt[n]{a^m}$. Applying this rule, we get:

$27^{\frac{2}{3}} = \sqrt[3]{27^2} = \sqrt[3]{729} = 9$

$64^{\frac{2}{3}} = \sqrt[3]{64^2} = \sqrt[3]{4096} = 16$

Simplifying the Expression

Now that we have evaluated the exponents, we can simplify the expression:

$\frac{1}{\frac{27^{\frac{2}{3}}}{64^{\frac{2}{3}}}} = \frac{1}{\frac{9}{16}} = \frac{16}{9}$

Conclusion

Therefore, the value of the expression $\left(\frac{27}{64}\right)^{-\frac{2}{3}}$ is $\frac{16}{9}$.

Comparison with Answer Choices

Comparing our result with the answer choices, we can see that the correct answer is:

• D. $\frac{16}{9}$

The other answer choices are incorrect:

• A. $-\frac{9}{16}$ is incorrect because the negative sign is not present in the original expression.
• B. $-\frac{9}{32}$ is incorrect because the denominator is not 32.
• C. $\frac{9}{32}$ is incorrect because the numerator is not 9.
• E. $\frac{32}{9}$ is incorrect because the denominator is not 9.

Final Answer

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

Understanding Negative Exponents

Negative exponents can be a bit tricky to work with, but with the right rules and techniques, you can simplify even the most complex expressions. In this article, we'll answer some common questions about evaluating expressions with negative exponents.

Q: What is the rule for negative exponents?

A: The rule for negative exponents states that $a^{-n} = \frac{1}{a^n}$. This means that when you see a negative exponent, you can rewrite the expression 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, follow these steps:

1. Rewrite the expression as a fraction with a positive exponent in the denominator using the rule $a^{-n} = \frac{1}{a^n}$.
2. Simplify the expression inside the parentheses, if necessary.
3. Evaluate any exponents that are present.
4. Simplify the resulting fraction, if necessary.

Q: What if I have a negative exponent with a fraction?

A: If you have a negative exponent with a fraction, you can rewrite the expression as a fraction with a positive exponent in the denominator using the rule $a^{-n} = \frac{1}{a^n}$. For example:

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

Q: How do I evaluate the exponent of a fraction?

A: To evaluate the exponent of a fraction, you can use the rule that states $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$. For example:

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

Q: What if I have a negative exponent with a variable?

A: If you have a negative exponent with a variable, you can rewrite the expression as a fraction with a positive exponent in the denominator using the rule $a^{-n} = \frac{1}{a^n}$. For example:

$x^{-2} = \frac{1}{x^2}$

Q: Can I simplify an expression with a negative exponent by canceling out terms?

A: Yes, you can simplify an expression with a negative exponent by canceling out terms. For example:

$\frac{1}{x^2} = \frac{1}{x \cdot x} = \frac{1}{x} \cdot \frac{1}{x} = \frac{1}{x^2}$

Q: What if I have a negative exponent with a radical?

A: If you have a negative exponent with a radical, you can rewrite the expression as a fraction with a positive exponent in the denominator using the rule $a^{-n} = \frac{1}{a^n}$. For example:

$\sqrt[3]{x}^{-2} = \frac{1}{\sqrt[3]{x^2}}$

Q: Can I use the rule for negative exponents to simplify an expression with a negative exponent and a radical?

A: Yes, you can use the rule for negative exponents to simplify an expression with a negative exponent and a radical. For example:

$\sqrt[3]{x}^{-2} = \frac{1}{\sqrt[3]{x^2}} = \frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x}} = \frac{1}{\sqrt[3]{x^2}}$

Conclusion

Evaluating expressions with negative exponents can be a bit tricky, but with the right rules and techniques, you can simplify even the most complex expressions. Remember to rewrite the expression as a fraction with a positive exponent in the denominator, simplify the expression inside the parentheses, evaluate any exponents that are present, and simplify the resulting fraction, if necessary.

Final Tips

• Always rewrite the expression as a fraction with a positive exponent in the denominator using the rule $a^{-n} = \frac{1}{a^n}$.
• Simplify the expression inside the parentheses, if necessary.
• Evaluate any exponents that are present.
• Simplify the resulting fraction, if necessary.
• Use the rule that states $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ to evaluate the exponent of a fraction.
• Use the rule that states $a^{\frac{m}{n}} = \sqrt[n]{a^m}$ to evaluate the exponent of a radical.

By following these tips and techniques, you can become a master of evaluating expressions with negative exponents!