Simplify And Evaluate $27^{-\frac{2}{3}}$.
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Introduction
When dealing with exponents and fractions, it's essential to understand the rules and properties that govern them. In this article, we will simplify and evaluate the expression $27^{-\frac{2}{3}}$. We will break down the problem step by step, using the properties of exponents and fractions to arrive at the final solution.
Understanding the Expression
The given expression is $27^{-\frac{2}{3}}$. To simplify this expression, we need to understand the properties of exponents and fractions. The exponent $-\frac{2}{3}$ indicates that we are dealing with a negative exponent, which can be rewritten as a positive exponent by taking the reciprocal of the base.
Rewriting the Expression
We can rewrite the expression $27^{-\frac{2}{3}}$ as $\frac{1}{27^{\frac{2}{3}}}$ by taking the reciprocal of the base. This is a fundamental property of exponents, which states that $a^{-n} = \frac{1}{a^n}$.
Simplifying the Expression
Now that we have rewritten the expression, we can simplify it further by evaluating the exponent $\frac{2}{3}$. To do this, we need to understand the concept of fractional exponents. A fractional exponent $\frac{m}{n}$ can be rewritten as $\sqrt[n]{x^m}$, where $x$ is the base.
Evaluating the Exponent
In this case, the exponent $\frac{2}{3}$ can be rewritten as $\sqrt[3]{x^2}$. Since the base is $27$, we can rewrite the expression as $\frac{1}{\sqrt[3]{27^2}}$.
Simplifying the Radical
Now that we have evaluated the exponent, we can simplify the radical $\sqrt[3]{27^2}$. To do this, we need to understand the concept of radicals and their properties. A radical $\sqrt[n]{x}$ can be rewritten as $x^{\frac{1}{n}}$.
Evaluating the Radical
In this case, the radical $\sqrt[3]{27^2}$ can be rewritten as $(272){\frac{1}{3}}$. Since the exponent $\frac{1}{3}$ is the reciprocal of the base $3$, we can simplify the expression further by taking the reciprocal of the base.
Simplifying the Expression
Now that we have evaluated the radical, we can simplify the expression further by taking the reciprocal of the base. This gives us $\frac{1}{(272){\frac{1}{3}}}$.
Evaluating the Expression
Finally, we can evaluate the expression $\frac{1}{(272){\frac{1}{3}}}$ by simplifying the exponent $\frac{1}{3}$. Since the base is $27^2$, we can rewrite the expression as $\frac{1}{(272){\frac{1}{3}}} = \frac{1}{(27{\frac{2}{3}}){\frac{1}{3}}}$.
Simplifying the Expression
Now that we have evaluated the expression, we can simplify it further by combining the exponents. This gives us $\frac{1}{27^{\frac{2}{3} \cdot \frac{1}{3}}}$.
Evaluating the Expression
Finally, we can evaluate the expression $\frac{1}{27^{\frac{2}{3} \cdot \frac{1}{3}}}$ by simplifying the exponent $\frac{2}{3} \cdot \frac{1}{3}$. This gives us $\frac{1}{27^{\frac{2}{9}}}$.
Simplifying the Expression
Now that we have evaluated the expression, we can simplify it further by rewriting the exponent $\frac{2}{9}$ as a radical. This gives us $\frac{1}{\sqrt[9]{27^2}}$.
Evaluating the Expression
Finally, we can evaluate the expression $\frac{1}{\sqrt[9]{27^2}}$ by simplifying the radical. This gives us $\frac{1}{\sqrt[9]{(33)2}}$.
Simplifying the Expression
Now that we have evaluated the expression, we can simplify it further by combining the exponents. This gives us $\frac{1}{\sqrt[9]{3^6}}$.
Evaluating the Expression
Finally, we can evaluate the expression $\frac{1}{\sqrt[9]{3^6}}$ by simplifying the radical. This gives us $\frac{1}{3^{\frac{6}{9}}}$.
Simplifying the Expression
Now that we have evaluated the expression, we can simplify it further by rewriting the exponent $\frac{6}{9}$ as a fraction. This gives us $\frac{1}{3^{\frac{2}{3}}}$.
Evaluating the Expression
Finally, we can evaluate the expression $\frac{1}{3^{\frac{2}{3}}}$ by simplifying the exponent. This gives us $\frac{1}{\sqrt[3]{3^2}}$.
Simplifying the Expression
Now that we have evaluated the expression, we can simplify it further by rewriting the radical $\sqrt[3]{3^2}$ as a fraction. This gives us $\frac{1}{\frac{3^2}{3}}$.
Evaluating the Expression
Finally, we can evaluate the expression $\frac{1}{\frac{3^2}{3}}$ by simplifying the fraction. This gives us $\frac{1}{\frac{9}{3}}$.
Simplifying the Expression
Now that we have evaluated the expression, we can simplify it further by rewriting the fraction $\frac{9}{3}$ as a decimal. This gives us $\frac{1}{3}$.
Conclusion
In conclusion, we have simplified and evaluated the expression $27^{-\frac{2}{3}}$ by using the properties of exponents and fractions. We have broken down the problem step by step, using the rules and properties of exponents and fractions to arrive at the final solution. The final answer is $\frac{1}{3}$.
Final Answer
The final answer is $\boxed{\frac{1}{3}}$.
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Introduction
In our previous article, we simplified and evaluated the expression $27^{-\frac{2}{3}}$. In this article, we will answer some common questions related to this topic.
Q: What is the rule for simplifying negative exponents?
A: The rule for simplifying negative exponents is to take the reciprocal of the base and change the sign of the exponent. In other words, $a^{-n} = \frac{1}{a^n}$.
Q: How do you rewrite a negative exponent as a positive exponent?
A: To rewrite a negative exponent as a positive exponent, you take the reciprocal of the base and change the sign of the exponent. For example, $27^{-\frac{2}{3}} = \frac{1}{27^{\frac{2}{3}}}$.
Q: What is the rule for simplifying fractional exponents?
A: The rule for simplifying fractional exponents is to rewrite the exponent as a radical. In other words, $a^{\frac{m}{n}} = \sqrt[n]{a^m}$.
Q: How do you simplify a radical with a fractional exponent?
A: To simplify a radical with a fractional exponent, you rewrite the exponent as a fraction and simplify the radical. For example, $\sqrt[3]{27^2} = (272){\frac{1}{3}}$.
Q: What is the rule for simplifying a radical with a fractional exponent?
A: The rule for simplifying a radical with a fractional exponent is to rewrite the exponent as a fraction and simplify the radical. In other words, $\sqrt[n]{a^m} = a^{\frac{m}{n}}$.
Q: How do you simplify a radical with a fractional exponent?
A: To simplify a radical with a fractional exponent, you rewrite the exponent as a fraction and simplify the radical. For example, $\sqrt[3]{27^2} = (272){\frac{1}{3}} = 27^{\frac{2}{3}}$.
Q: What is the final answer to the expression $27^{-\frac{2}{3}}$?
A: The final answer to the expression $27^{-\frac{2}{3}}$ is $\frac{1}{3}$.
Q: How do you evaluate an expression with a negative exponent?
A: To evaluate an expression with a negative exponent, you take the reciprocal of the base and change the sign of the exponent. For example, $27^{-\frac{2}{3}} = \frac{1}{27^{\frac{2}{3}}}$.
Q: What is the rule for simplifying a fraction with a negative exponent?
A: The rule for simplifying a fraction with a negative exponent is to take the reciprocal of the base and change the sign of the exponent. In other words, $\frac{a{-n}}{bn} = \frac{bn}{an}$.
Q: How do you simplify a fraction with a negative exponent?
A: To simplify a fraction with a negative exponent, you take the reciprocal of the base and change the sign of the exponent. For example, $\frac{27{-\frac{2}{3}}}{3{\frac{2}{3}}} = \frac{3{\frac{2}{3}}}{27{\frac{2}{3}}}$.
Q: What is the final answer to the expression $\frac{27{-\frac{2}{3}}}{3{\frac{2}{3}}}$?
A: The final answer to the expression $\frac{27{-\frac{2}{3}}}{3{\frac{2}{3}}}$ is $\frac{1}{3}$.
Conclusion
In conclusion, we have answered some common questions related to simplifying and evaluating the expression $27^{-\frac{2}{3}}$. We have covered topics such as simplifying negative exponents, rewriting negative exponents as positive exponents, simplifying fractional exponents, and evaluating expressions with negative exponents.
Final Answer
The final answer is $\boxed{\frac{1}{3}}$.