Evaluate The Expression: ( 27 8 ) 2 3 \left(\frac{27}{8}\right)^{\frac{2}{3}} ( 8 27 ​ ) 3 2 ​

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Introduction

In mathematics, expressions involving exponents and fractions can be challenging to evaluate. The given expression, $\left(\frac{27}{8}\right)^{\frac{2}{3}}$, is a perfect example of such a problem. In this article, we will break down the expression and provide a step-by-step solution to evaluate it.

Understanding the Expression

The given expression involves a fraction, $\frac{27}{8}$, raised to the power of $\frac{2}{3}$. To evaluate this expression, we need to understand the concept of fractional exponents and how to simplify them.

Fractional Exponents

Fractional exponents are a way of expressing a number raised to a power that is a fraction. The general form of a fractional exponent is $a^{\frac{m}{n}}$, where $a$ is the base, $m$ is the numerator, and $n$ is the denominator.

In the given expression, the base is $\frac{27}{8}$, the numerator is $2$, and the denominator is $3$. Therefore, the expression can be written as $\left(\frac{27}{8}\right)^{\frac{2}{3}}$.

Simplifying the Expression

To simplify the expression, we need to apply the rules of exponents. The rule states that $(a^m)^n = a^{mn}$. In this case, we can rewrite the expression as $\left(\frac{27}{8}\right)^{\frac{2}{3}} = \left(\frac{27}{8}\right)^{\frac{1}{3} \cdot 2}$.

Applying the Rule of Exponents

Now, we can apply the rule of exponents to simplify the expression. We have $\left(\frac{27}{8}\right)^{\frac{1}{3} \cdot 2} = \left(\frac{27}{8}\right)^{\frac{1}{3}} \cdot \left(\frac{27}{8}\right)^{\frac{1}{3}} \cdot \left(\frac{27}{8}\right)^{\frac{1}{3}}$.

Simplifying the Expression Further

We can simplify the expression further by evaluating the cube root of the fraction. The cube root of a fraction can be evaluated by taking the cube root of the numerator and the denominator separately.

$\left(\frac{27}{8}\right)^{\frac{1}{3}} = \frac{\sqrt[3]{27}}{\sqrt[3]{8}}$

Evaluating the Cube Root

Now, we can evaluate the cube root of the numerator and the denominator.

$\sqrt[3]{27} = 3$

$\sqrt[3]{8} = 2$

Simplifying the Expression

Now, we can simplify the expression by substituting the values of the cube root.

$\frac{\sqrt[3]{27}}{\sqrt[3]{8}} = \frac{3}{2}$

Evaluating the Expression

Finally, we can evaluate the expression by multiplying the three cube roots together.

$\left(\frac{27}{8}\right)^{\frac{1}{3}} \cdot \left(\frac{27}{8}\right)^{\frac{1}{3}} \cdot \left(\frac{27}{8}\right)^{\frac{1}{3}} = \frac{3}{2} \cdot \frac{3}{2} \cdot \frac{3}{2}$

Simplifying the Expression

Now, we can simplify the expression by multiplying the fractions together.

$\frac{3}{2} \cdot \frac{3}{2} \cdot \frac{3}{2} = \frac{27}{8}$

Evaluating the Expression

Finally, we can evaluate the expression by simplifying the fraction.

$\frac{27}{8} = \boxed{3.375}$

Conclusion

In this article, we evaluated the expression $\left(\frac{27}{8}\right)^{\frac{2}{3}}$ using the rules of exponents and fractional exponents. We simplified the expression by applying the rule of exponents and evaluating the cube root of the fraction. Finally, we evaluated the expression by multiplying the three cube roots together and simplifying the fraction.

Frequently Asked Questions

• What is the value of $\left(\frac{27}{8}\right)^{\frac{2}{3}}$?
• How do you evaluate an expression with a fractional exponent?
• What is the rule of exponents for fractional exponents?

References

• [1] "Exponents and Exponential Functions" by Math Open Reference
• [2] "Fractional Exponents" by Khan Academy
• [3] "Rules of Exponents" by Purplemath

Further Reading

• "Exponents and Exponential Functions" by Math Open Reference
• "Fractional Exponents" by Khan Academy
• "Rules of Exponents" by Purplemath
  

Introduction

In our previous article, we evaluated the expression $\left(\frac{27}{8}\right)^{\frac{2}{3}}$ using the rules of exponents and fractional exponents. In this article, we will answer some frequently asked questions related to evaluating expressions with fractional exponents.

Q&A

Q: What is the value of $\left(\frac{27}{8}\right)^{\frac{2}{3}}$?

A: The value of $\left(\frac{27}{8}\right)^{\frac{2}{3}}$ is $\boxed{3.375}$.

Q: How do you evaluate an expression with a fractional exponent?

A: To evaluate an expression with a fractional exponent, you need to apply the rule of exponents, which states that $(a^m)^n = a^{mn}$. You can also use the property of fractional exponents, which states that $a^{\frac{m}{n}} = \sqrt[n]{a^m}$.

Q: What is the rule of exponents for fractional exponents?

A: The rule of exponents for fractional exponents is $(a^m)^n = a^{mn}$. This rule can be used to simplify expressions with fractional exponents.

Q: How do you simplify an expression with a fractional exponent?

A: To simplify an expression with a fractional exponent, you need to apply the rule of exponents and the property of fractional exponents. You can also use the fact that $a^{\frac{m}{n}} = \sqrt[n]{a^m}$.

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

A: A fractional exponent is an exponent that is a fraction, while a negative exponent is an exponent that is negative. For example, $a^{\frac{1}{2}}$ is a fractional exponent, while $a^{-1}$ is a negative exponent.

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

A: To evaluate an expression with a negative exponent, you need to apply the rule of exponents, which states that $a^{-m} = \frac{1}{a^m}$.

Q: What is the property of fractional exponents?

A: The property of fractional exponents states that $a^{\frac{m}{n}} = \sqrt[n]{a^m}$. This property can be used to simplify expressions with fractional exponents.

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

A: To simplify an expression with a cube root, you need to apply the property of cube roots, which states that $\sqrt[3]{a^m} = a^{\frac{m}{3}}$.

Conclusion

In this article, we answered some frequently asked questions related to evaluating expressions with fractional exponents. We also provided some examples and explanations to help you understand the concepts.

Frequently Asked Questions

• What is the value of $\left(\frac{27}{8}\right)^{\frac{2}{3}}$?
• How do you evaluate an expression with a fractional exponent?
• What is the rule of exponents for fractional exponents?
• How do you simplify an expression with a fractional exponent?
• What is the difference between a fractional exponent and a negative exponent?
• How do you evaluate an expression with a negative exponent?
• What is the property of fractional exponents?
• How do you simplify an expression with a cube root?

References

• [1] "Exponents and Exponential Functions" by Math Open Reference
• [2] "Fractional Exponents" by Khan Academy
• [3] "Rules of Exponents" by Purplemath

Further Reading

• "Exponents and Exponential Functions" by Math Open Reference
• "Fractional Exponents" by Khan Academy
• "Rules of Exponents" by Purplemath