Rewrite The Following Without An Exponent. ( 4 7 ) − 2 \left(\frac{4}{7}\right)^{-2} ( 7 4 ​ ) − 2

Understanding Exponents and Fractions

Exponents and fractions are two fundamental concepts in mathematics that are often used together in various mathematical operations. In this article, we will focus on rewriting exponents in fractions, specifically the expression $\left(\frac{4}{7}\right)^{-2}$.

What is an Exponent?

An exponent is a small number that is written to the upper right of a number or a variable. It represents the power to which the base number or variable is raised. For example, in the expression $2^3$, the exponent $3$ indicates that the base number $2$ is raised to the power of $3$. In other words, $2^3 = 2 \times 2 \times 2 = 8$.

Rewriting Exponents in Fractions

When an exponent is applied to a fraction, it can be rewritten in a different form using the properties of exponents. The expression $\left(\frac{4}{7}\right)^{-2}$ can be rewritten using the property of negative exponents.

Property of Negative Exponents

The property of negative exponents states that for any non-zero number $a$ and any integer $n$, $a^{-n} = \frac{1}{a^n}$. This property can be used to rewrite the expression $\left(\frac{4}{7}\right)^{-2}$.

Rewriting the Expression

Using the property of negative exponents, we can rewrite the expression $\left(\frac{4}{7}\right)^{-2}$ as follows:

$$\left(\frac{4}{7}\right)^{-2} = \frac{1}{\left(\frac{4}{7}\right)^2}$$

Simplifying the Expression

To simplify the expression, we can evaluate the exponent in the denominator.

$$\left(\frac{4}{7}\right)^2 = \frac{4^2}{7^2} = \frac{16}{49}$$

Final Result

Substituting the simplified expression back into the original expression, we get:

$$\left(\frac{4}{7}\right)^{-2} = \frac{1}{\frac{16}{49}} = \frac{49}{16}$$

Conclusion

In this article, we have rewritten the expression $\left(\frac{4}{7}\right)^{-2}$ using the property of negative exponents. We have also simplified the expression to obtain the final result. This demonstrates the importance of understanding the properties of exponents and fractions in mathematics.

Common Mistakes to Avoid

When rewriting exponents in fractions, it is essential to remember the following common mistakes to avoid:

• Incorrect application of the property of negative exponents: Make sure to apply the property correctly to avoid errors.
• Insufficient simplification: Ensure that the expression is fully simplified to obtain the correct result.
• Failure to check for zero denominators: Be cautious when working with fractions and check for zero denominators to avoid division by zero.

Real-World Applications

Rewriting exponents in fractions has numerous real-world applications in various fields, including:

• Science: Exponents and fractions are used to describe the behavior of physical systems, such as chemical reactions and population growth.
• Engineering: Exponents and fractions are used to design and optimize systems, such as electrical circuits and mechanical systems.
• Finance: Exponents and fractions are used to calculate interest rates and investment returns.

Practice Problems

To reinforce your understanding of rewriting exponents in fractions, try the following practice problems:

1. Rewrite the expression $\left(\frac{3}{5}\right)^{-3}$ using the property of negative exponents.
2. Simplify the expression $\left(\frac{2}{3}\right)^{-2}$.
3. Evaluate the expression $\left(\frac{4}{9}\right)^{-1}$.

Conclusion

Understanding Exponents and Fractions

Exponents and fractions are two fundamental concepts in mathematics that are often used together in various mathematical operations. In this article, we will focus on rewriting exponents in fractions, specifically the expression $\left(\frac{4}{7}\right)^{-2}$.

Q&A: Rewriting Exponents in Fractions

Q: What is an exponent?

A: An exponent is a small number that is written to the upper right of a number or a variable. It represents the power to which the base number or variable is raised.

Q: How do I rewrite an exponent in a fraction?

A: To rewrite an exponent in a fraction, you can use the property of negative exponents. The property states that for any non-zero number $a$ and any integer $n$, $a^{-n} = \frac{1}{a^n}$.

Q: What is the property of negative exponents?

A: The property of negative exponents states that for any non-zero number $a$ and any integer $n$, $a^{-n} = \frac{1}{a^n}$. This property can be used to rewrite the expression $\left(\frac{4}{7}\right)^{-2}$.

Q: How do I simplify the expression $\left(\frac{4}{7}\right)^{-2}$?

A: To simplify the expression, you can evaluate the exponent in the denominator. $\left(\frac{4}{7}\right)^2 = \frac{4^2}{7^2} = \frac{16}{49}$

Q: What is the final result of the expression $\left(\frac{4}{7}\right)^{-2}$?

A: The final result of the expression $\left(\frac{4}{7}\right)^{-2}$ is $\frac{49}{16}$.

Q: What are some common mistakes to avoid when rewriting exponents in fractions?

A: Some common mistakes to avoid when rewriting exponents in fractions include:

• Incorrect application of the property of negative exponents: Make sure to apply the property correctly to avoid errors.
• Insufficient simplification: Ensure that the expression is fully simplified to obtain the correct result.
• Failure to check for zero denominators: Be cautious when working with fractions and check for zero denominators to avoid division by zero.

Q: What are some real-world applications of rewriting exponents in fractions?

A: Rewriting exponents in fractions has numerous real-world applications in various fields, including:

• Science: Exponents and fractions are used to describe the behavior of physical systems, such as chemical reactions and population growth.
• Engineering: Exponents and fractions are used to design and optimize systems, such as electrical circuits and mechanical systems.
• Finance: Exponents and fractions are used to calculate interest rates and investment returns.

Q: How can I practice rewriting exponents in fractions?

A: You can practice rewriting exponents in fractions by trying the following practice problems:

1. Rewrite the expression $\left(\frac{3}{5}\right)^{-3}$ using the property of negative exponents.
2. Simplify the expression $\left(\frac{2}{3}\right)^{-2}$.
3. Evaluate the expression $\left(\frac{4}{9}\right)^{-1}$.

Conclusion

In conclusion, rewriting exponents in fractions is a fundamental concept in mathematics that has numerous real-world applications. By understanding the properties of exponents and fractions, you can simplify complex expressions and solve problems in various fields. Remember to avoid common mistakes and practice regularly to reinforce your understanding.

Additional Resources

For more information on rewriting exponents in fractions, check out the following resources:

• Math textbooks: Consult a math textbook for a comprehensive explanation of exponents and fractions.
• Online resources: Visit online resources such as Khan Academy, Mathway, and Wolfram Alpha for interactive lessons and practice problems.
• Math tutors: Consider hiring a math tutor to provide one-on-one instruction and guidance.

Final Tips

• Practice regularly: Regular practice will help you develop a strong understanding of rewriting exponents in fractions.
• Use online resources: Online resources such as Khan Academy, Mathway, and Wolfram Alpha can provide interactive lessons and practice problems to help you learn.
• Seek help when needed: Don't hesitate to seek help from a math tutor or online resource if you're struggling with a concept.