Evaluate. Write Your Answers As Fractions.$\[ \begin{array}{l} \left(-\frac{5}{2}\right)^2= \\ -\left(\frac{4}{5}\right)^4= \end{array} \\]

Introduction

In mathematics, fractions are a fundamental concept that plays a crucial role in various mathematical operations. When dealing with fractions, it's essential to understand the rules and properties that govern their behavior. In this article, we will explore the evaluation of expressions involving fractions, focusing on the use of exponents and the properties of negative numbers.

Evaluating the First Expression

The first expression we need to evaluate is $\left(-\frac{5}{2}\right)^2$. To do this, we need to apply the exponent rule, which states that $(a^m)^n = a^{m \cdot n}$.

\left(-\frac{5}{2}\right)^2 = \left(\frac{5}{2}\right)^2

Now, we can simplify the expression by squaring the fraction.

\left(\frac{5}{2}\right)^2 = \frac{5^2}{2^2} = \frac{25}{4}

Therefore, the value of the first expression is $\frac{25}{4}$.

Evaluating the Second Expression

The second expression we need to evaluate is $-\left(\frac{4}{5}\right)^4$. To do this, we need to apply the exponent rule and the property of negative numbers.

-\left(\frac{4}{5}\right)^4 = -\frac{4^4}{5^4}

Now, we can simplify the expression by evaluating the exponent.

-\frac{4^4}{5^4} = -\frac{256}{625}

Therefore, the value of the second expression is $-\frac{256}{625}$.

Discussion and Conclusion

In this article, we evaluated two expressions involving fractions, focusing on the use of exponents and the properties of negative numbers. We applied the exponent rule and the property of negative numbers to simplify the expressions and arrive at their final values.

Key Takeaways

• When evaluating expressions involving fractions, it's essential to apply the exponent rule and the property of negative numbers.
• The exponent rule states that $(a^m)^n = a^{m \cdot n}$.
• The property of negative numbers states that $(-a)^n = a^n$ if $n$ is even and $(-a)^n = -a^n$ if $n$ is odd.

Final Thoughts

Evaluating expressions involving fractions requires a deep understanding of the rules and properties that govern their behavior. By applying the exponent rule and the property of negative numbers, we can simplify complex expressions and arrive at their final values. This article has provided a comprehensive overview of the evaluation of expressions involving fractions, and we hope that it has been informative and helpful.

Additional Resources

For further reading on the evaluation of expressions involving fractions, we recommend the following resources:

• Khan Academy: Exponents and Exponential Functions
• Mathway: Exponents and Exponential Functions
• Wolfram Alpha: Exponents and Exponential Functions

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "Mathematics for the Nonmathematician" by Morris Kline
• [3] "Calculus" by Michael Spivak
  Evaluating Expressions with Fractions: A Q&A Guide =====================================================

Introduction

In our previous article, we explored the evaluation of expressions involving fractions, focusing on the use of exponents and the properties of negative numbers. In this article, we will provide a Q&A guide to help you better understand the concepts and rules involved in evaluating expressions with fractions.

Q&A: Evaluating Expressions with Fractions

Q: What is the exponent rule for fractions?

A: The exponent rule for fractions states that $(a^m)^n = a^{m \cdot n}$. This means that when you raise a fraction to a power, you can multiply the exponent by the power.

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

A: To simplify an expression with a negative exponent, you can rewrite the expression with a positive exponent by taking the reciprocal of the fraction. For example, $\frac{1}{a^m} = a^{-m}$.

Q: What is the property of negative numbers for exponents?

A: The property of negative numbers for exponents states that $(-a)^n = a^n$ if $n$ is even and $(-a)^n = -a^n$ if $n$ is odd. This means that when you raise a negative number to an even power, the result is positive, and when you raise a negative number to an odd power, the result is negative.

Q: How do I evaluate an expression with multiple exponents?

A: To evaluate an expression with multiple exponents, you can apply the exponent rule and the property of negative numbers. For example, $\left(-\frac{5}{2}\right)^2 = \left(\frac{5}{2}\right)^2 = \frac{25}{4}$.

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

A: A positive exponent indicates that the base is raised to a power, while a negative exponent indicates that the reciprocal of the base is raised to a power. For example, $a^m$ is different from $a^{-m}$.

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

A: To simplify an expression with a fraction and an exponent, you can apply the exponent rule and the property of negative numbers. For example, $-\left(\frac{4}{5}\right)^4 = -\frac{4^4}{5^4} = -\frac{256}{625}$.

Common Mistakes to Avoid

When evaluating expressions with fractions, it's essential to avoid common mistakes. Here are a few to watch out for:

• Forgetting to apply the exponent rule: Make sure to apply the exponent rule when evaluating expressions with fractions.
• Not simplifying the expression: Simplify the expression by applying the exponent rule and the property of negative numbers.
• Not considering the sign of the exponent: Consider the sign of the exponent when evaluating expressions with fractions.

Conclusion

Evaluating expressions with fractions requires a deep understanding of the rules and properties that govern their behavior. By applying the exponent rule and the property of negative numbers, you can simplify complex expressions and arrive at their final values. This Q&A guide has provided a comprehensive overview of the evaluation of expressions with fractions, and we hope that it has been informative and helpful.

Additional Resources

For further reading on the evaluation of expressions with fractions, we recommend the following resources:

• Khan Academy: Exponents and Exponential Functions
• Mathway: Exponents and Exponential Functions
• Wolfram Alpha: Exponents and Exponential Functions

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "Mathematics for the Nonmathematician" by Morris Kline
• [3] "Calculus" by Michael Spivak