Simplify The Expression: { \left(\frac{3}{5}\right)^{-2}$}$

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Understanding the Concept of Negative Exponents

In mathematics, a negative exponent is a shorthand way of expressing a fraction. When we see a negative exponent, it means that we need to take the reciprocal of the base and change the sign of the exponent. In this case, we have $\left(\frac{3}{5}\right)^{-2}$, which can be rewritten as $\frac{1}{\left(\frac{3}{5}\right)^2}$.

Recall the Rule for Negative Exponents

The rule for negative exponents states that $a^{-n} = \frac{1}{a^n}$. This means that when we see a negative exponent, we can rewrite it as a fraction with the reciprocal of the base and the opposite sign of the exponent.

Applying the Rule to the Given Expression

Using the rule for negative exponents, we can rewrite the given expression as $\frac{1}{\left(\frac{3}{5}\right)^2}$. To simplify this expression, we need to evaluate the exponent.

Evaluating the Exponent

To evaluate the exponent, we need to square the fraction $\frac{3}{5}$. This means that we need to multiply the numerator and denominator by themselves.

$\left(\frac{3}{5}\right)^2 = \frac{3 \cdot 3}{5 \cdot 5} = \frac{9}{25}$

Substituting the Result Back into the Expression

Now that we have evaluated the exponent, we can substitute the result back into the expression.

$\frac{1}{\left(\frac{3}{5}\right)^2} = \frac{1}{\frac{9}{25}}$

Simplifying the Expression

To simplify the expression, we need to get rid of the fraction in the denominator. We can do this by multiplying the numerator and denominator by the reciprocal of the fraction in the denominator.

$\frac{1}{\frac{9}{25}} = 1 \cdot \frac{25}{9} = \frac{25}{9}$

Conclusion

In this article, we have simplified the expression $\left(\frac{3}{5}\right)^{-2}$. We used the rule for negative exponents to rewrite the expression as a fraction with the reciprocal of the base and the opposite sign of the exponent. We then evaluated the exponent and substituted the result back into the expression. Finally, we simplified the expression to get the final answer.

Key Takeaways

• Negative exponents can be rewritten as fractions with the reciprocal of the base and the opposite sign of the exponent.
• To evaluate an exponent, we need to multiply the base by itself as many times as the exponent indicates.
• To simplify an expression with a negative exponent, we need to get rid of the fraction in the denominator by multiplying the numerator and denominator by the reciprocal of the fraction in the denominator.

Real-World Applications

Negative exponents have many real-world applications in fields such as physics, engineering, and economics. For example, in physics, negative exponents are used to describe the behavior of particles at very small distances or high energies. In engineering, negative exponents are used to describe the behavior of systems with negative feedback. In economics, negative exponents are used to describe the behavior of markets with negative feedback loops.

Common Mistakes to Avoid

When working with negative exponents, there are several common mistakes to avoid. These include:

• Forgetting to change the sign of the exponent when rewriting a negative exponent as a fraction.
• Forgetting to evaluate the exponent when simplifying an expression with a negative exponent.
• Forgetting to get rid of the fraction in the denominator when simplifying an expression with a negative exponent.

Practice Problems

To practice simplifying expressions with negative exponents, try the following problems:

• Simplify the expression $\left(\frac{2}{3}\right)^{-3}$.
• Simplify the expression $\left(\frac{4}{5}\right)^{-2}$.
• Simplify the expression $\left(\frac{3}{2}\right)^{-4}$.

Conclusion

Q: What is a negative exponent?

A: A negative exponent is a shorthand way of expressing a fraction. When we see a negative exponent, it means that we need to take the reciprocal of the base and change the sign of the exponent.

Q: How do I rewrite a negative exponent as a fraction?

A: To rewrite a negative exponent as a fraction, we need to take the reciprocal of the base and change the sign of the exponent. For example, $\left(\frac{3}{5}\right)^{-2}$ can be rewritten as $\frac{1}{\left(\frac{3}{5}\right)^2}$.

Q: How do I evaluate an exponent?

A: To evaluate an exponent, we need to multiply the base by itself as many times as the exponent indicates. For example, $\left(\frac{3}{5}\right)^2$ means that we need to multiply $\frac{3}{5}$ by itself.

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

A: To simplify an expression with a negative exponent, we need to get rid of the fraction in the denominator by multiplying the numerator and denominator by the reciprocal of the fraction in the denominator. For example, $\frac{1}{\left(\frac{3}{5}\right)^2}$ can be simplified to $\frac{25}{9}$.

Q: What are some common mistakes to avoid when working with negative exponents?

A: Some common mistakes to avoid when working with negative exponents include:

• Forgetting to change the sign of the exponent when rewriting a negative exponent as a fraction.
• Forgetting to evaluate the exponent when simplifying an expression with a negative exponent.
• Forgetting to get rid of the fraction in the denominator when simplifying an expression with a negative exponent.

Q: How do I apply the concept of negative exponents to real-world problems?

A: Negative exponents have many real-world applications in fields such as physics, engineering, and economics. For example, in physics, negative exponents are used to describe the behavior of particles at very small distances or high energies. In engineering, negative exponents are used to describe the behavior of systems with negative feedback. In economics, negative exponents are used to describe the behavior of markets with negative feedback loops.

Q: What are some practice problems that I can use to practice simplifying expressions with negative exponents?

A: Here are some practice problems that you can use to practice simplifying expressions with negative exponents:

• Simplify the expression $\left(\frac{2}{3}\right)^{-3}$.
• Simplify the expression $\left(\frac{4}{5}\right)^{-2}$.
• Simplify the expression $\left(\frac{3}{2}\right)^{-4}$.

Q: How do I know if I have simplified an expression with a negative exponent correctly?

A: To check if you have simplified an expression with a negative exponent correctly, you can plug in some values for the base and exponent and see if the result is correct. You can also use a calculator to check your answer.

Q: What are some tips for simplifying expressions with negative exponents?

A: Here are some tips for simplifying expressions with negative exponents:

• Make sure to change the sign of the exponent when rewriting a negative exponent as a fraction.
• Make sure to evaluate the exponent when simplifying an expression with a negative exponent.
• Make sure to get rid of the fraction in the denominator when simplifying an expression with a negative exponent.
• Use a calculator to check your answer.

Conclusion

In conclusion, simplifying expressions with negative exponents requires a good understanding of the rule for negative exponents and the ability to evaluate exponents. By following the steps outlined in this article and practicing with the practice problems, you can simplify expressions with negative exponents and apply the concepts to real-world problems.