Exponentiation and Negative Exponents

Oct 13, 2025 by pynkocean.com 26 views

Understanding the Problem

When evaluating the expression ${$ (0.04)^{-1.5}$}$, we need to follow the order of operations, which is Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). In this case, we have an exponentiation operation that needs to be evaluated first.

Exponentiation and Negative Exponents

To evaluate the expression ${$ (0.04)^{-1.5}$}$, we need to understand the concept of negative exponents. A negative exponent indicates that we need to take the reciprocal of the base and change the sign of the exponent. In this case, we have ${$ (0.04)^{-1.5}$}$, which can be rewritten as ${$ \frac{1}{(0.04)^{1.5}}$}$.

Evaluating the Expression

Now, let's evaluate the expression ${$ \frac{1}{(0.04)^{1.5}}$}$. To do this, we need to follow the order of operations and evaluate the exponentiation operation first.

Exponentiation of 0.04

To evaluate the expression ${$ (0.04)^{1.5}$}$, we can use the fact that ${$ a^{m+n} = a^m \cdot a^n$}$. In this case, we have ${$ (0.04)^{1.5} = (0.04)^1 \cdot (0.04)^{0.5}$].

Evaluating the Square Root

To evaluate the expression [ $(0.04)^{0.5}\$}$ , we need to find the square root of 0.04. The square root of a number is a value that, when multiplied by itself, gives the original number. In this case, we have ${$ \sqrt{0.04} = \sqrt{\frac{4}{100}} = \frac{\sqrt{4}}{\sqrt{100}} = \frac{2}{10} = 0.2$}$.

Evaluating the Expression

Now, let's evaluate the expression ${$ (0.04)^{1.5} = (0.04)^1 \cdot (0.04)^{0.5} = 0.04 \cdot 0.2 = 0.008$}$.

Evaluating the Reciprocal

Now, let's evaluate the reciprocal of the expression ${$ (0.04)^{1.5}$}$. The reciprocal of a number is 1 divided by the number. In this case, we have ${$ \frac{1}{(0.04)^{1.5}} = \frac{1}{0.008} = 125$}$.

Conclusion

In conclusion, the value of the expression ${$ (0.04)^{-1.5}$}$ is 125.

Final Answer

The final answer is B. 125.

Frequently Asked Questions

Q: What is the order of operations in mathematics?

A: The order of operations in mathematics is Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

Q: What is the meaning of a negative exponent?

A: A negative exponent indicates that we need to take the reciprocal of the base and change the sign of the exponent.

Q: How do we evaluate the expression ${$ (0.04)^{-1.5}$}$?

A: To evaluate the expression ${$ (0.04)^{-1.5}$}$, we need to follow the order of operations and evaluate the exponentiation operation first. We can rewrite the expression as ${$ \frac{1}{(0.04)^{1.5}}$}$ and then evaluate the exponentiation operation.

Q: How do we evaluate the expression ${$ (0.04)^{1.5}$}$?

A: To evaluate the expression ${$ (0.04)^{1.5}$}$, we can use the fact that ${$ a^{m+n} = a^m \cdot a^n$}$. In this case, we have ${$ (0.04)^{1.5} = (0.04)^1 \cdot (0.04)^{0.5}$].

Q: How do we evaluate the square root of 0.04?

A: To evaluate the square root of 0.04, we need to find the value that, when multiplied by itself, gives the original number. In this case, we have [ $\sqrt{0.04} = \sqrt{\frac{4}{100}} = \frac{\sqrt{4}}{\sqrt{100}} = \frac{2}{10} = 0.2\$}$ .

Q: What is the final answer to the expression ${$ (0.04)^{-1.5}$}$?

A: The final answer to the expression ${$ (0.04)^{-1.5}$}$ is 125.

Q: Why is the final answer 125?

A: The final answer is 125 because we evaluated the expression ${$ (0.04)^{-1.5}$}$ by following the order of operations and taking the reciprocal of the base and changing the sign of the exponent.

Common Mistakes

Mistake 1: Not following the order of operations

A: Not following the order of operations can lead to incorrect results. In this case, we need to evaluate the exponentiation operation first and then take the reciprocal of the base and change the sign of the exponent.

Mistake 2: Not understanding the concept of negative exponents

A: Not understanding the concept of negative exponents can lead to incorrect results. In this case, we need to take the reciprocal of the base and change the sign of the exponent.

Tips and Tricks

Tip 1: Use the order of operations to evaluate expressions

A: Using the order of operations can help us evaluate expressions correctly and avoid mistakes.

Tip 2: Understand the concept of negative exponents

A: Understanding the concept of negative exponents can help us evaluate expressions correctly and avoid mistakes.

Conclusion

In conclusion, evaluating the expression ${$ (0.04)^{-1.5}$}$ requires us to follow the order of operations and understand the concept of negative exponents. By following these steps, we can arrive at the correct answer, which is 125.