Lola Says These Two Expressions Have The Same Value.Expression A:${ \left[\left(\frac{a}{b}\right) {-4}\right] 0 }$Expression B: (missing Expression)Which Explains Whether Lola Is Correct?A. Lola Is Correct Because Each Expression Has A

Introduction

In mathematics, expressions are used to represent values or relationships between variables. When evaluating expressions, it's essential to follow the correct order of operations to ensure accurate results. In this article, we'll examine two expressions, A and B, and determine whether they have the same value.

Expression A

The first expression is given as:

$$\left[\left(\frac{a}{b}\right)^{-4}\right]^0$$

This expression involves exponentiation and a power of zero. To evaluate it, 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).

Step 1: Evaluate the innermost parentheses

The innermost parentheses contain the fraction $\frac{a}{b}$ raised to the power of $-4$. This can be rewritten as:

$$\left(\frac{a}{b}\right)^{-4} = \frac{1}{\left(\frac{a}{b}\right)^4}$$

Step 2: Evaluate the exponent

The exponent is $-4$, which means we need to take the reciprocal of the expression inside the parentheses. This results in:

$$\frac{1}{\left(\frac{a}{b}\right)^4} = \frac{b^4}{a^4}$$

Step 3: Evaluate the outermost parentheses

The outermost parentheses contain the expression $\frac{b^4}{a^4}$ raised to the power of $0$. Any non-zero number raised to the power of $0$ is equal to $1$. Therefore, we have:

$$\left(\frac{b^4}{a^4}\right)^0 = 1$$

Expression B

The second expression is missing, but we can infer its value based on the given information. Since Lola claims that the two expressions have the same value, we can assume that Expression B is:

$$\left(\frac{a}{b}\right)^{-4}$$

Step 1: Evaluate the expression

This expression involves exponentiation. To evaluate it, we need to follow the order of operations. The exponent is $-4$, which means we need to take the reciprocal of the expression inside the parentheses. This results in:

$$\left(\frac{a}{b}\right)^{-4} = \frac{b^4}{a^4}$$

Comparison of Expressions A and B

Now that we have evaluated both expressions, we can compare their values. Expression A is equal to $1$, while Expression B is equal to $\frac{b^4}{a^4}$. Since these two expressions do not have the same value, Lola is incorrect.

Conclusion

In conclusion, we have analyzed two expressions, A and B, and determined that they do not have the same value. Expression A is equal to $1$, while Expression B is equal to $\frac{b^4}{a^4}$. This demonstrates the importance of following the correct order of operations when evaluating mathematical expressions.

Discussion

This problem highlights the need for careful attention to detail when working with mathematical expressions. Even small mistakes can lead to incorrect results. By following the order of operations and carefully evaluating each step, we can ensure accurate results and avoid errors.

Additional Examples

To further illustrate the concept, let's consider a few additional examples:

• Expression C: $\left(\frac{a}{b}\right)^{-3}$
• Expression D: $\left(\frac{a}{b}\right)^{-2}$

Using the same steps as before, we can evaluate these expressions and compare their values.

Expression C

The expression is given as:

$$\left(\frac{a}{b}\right)^{-3}$$

Following the order of operations, we can evaluate this expression as:

$$\left(\frac{a}{b}\right)^{-3} = \frac{b^3}{a^3}$$

Expression D

The expression is given as:

$$\left(\frac{a}{b}\right)^{-2}$$

Following the order of operations, we can evaluate this expression as:

$$\left(\frac{a}{b}\right)^{-2} = \frac{b^2}{a^2}$$

Comparison of Expressions C and D

Now that we have evaluated both expressions, we can compare their values. Expression C is equal to $\frac{b^3}{a^3}$, while Expression D is equal to $\frac{b^2}{a^2}$. Since these two expressions do not have the same value, we can conclude that Lola is incorrect.

Conclusion

In conclusion, we have analyzed three expressions, A, B, and C, and determined that they do not have the same value. Expression A is equal to $1$, while Expressions B and C are equal to $\frac{b^4}{a^4}$ and $\frac{b^3}{a^3}$, respectively. This demonstrates the importance of following the correct order of operations when evaluating mathematical expressions.

Final Thoughts

Q: What is the correct order of operations when evaluating mathematical expressions?

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

Q: How do you evaluate an expression with a power of zero?

A: When an expression is raised to the power of zero, the result is always 1, unless the expression is equal to zero.

Q: What is the value of the expression $\left(\frac{a}{b}\right)^{-4}$?

A: The value of the expression $\left(\frac{a}{b}\right)^{-4}$ is $\frac{b^4}{a^4}$.

Q: How do you compare the values of two expressions?

A: To compare the values of two expressions, you need to evaluate each expression separately and then compare their results.

Q: What is the value of the expression $\left(\frac{a}{b}\right)^{-3}$?

A: The value of the expression $\left(\frac{a}{b}\right)^{-3}$ is $\frac{b^3}{a^3}$.

Q: What is the value of the expression $\left(\frac{a}{b}\right)^{-2}$?

A: The value of the expression $\left(\frac{a}{b}\right)^{-2}$ is $\frac{b^2}{a^2}$.

Q: Can you provide an example of how to evaluate an expression with multiple exponents?

A: Yes, here's an example:

Suppose we have the expression $\left(\frac{a}{b}\right)^{-4} \cdot \left(\frac{a}{b}\right)^{-2}$.

To evaluate this expression, we need to follow the order of operations. First, we need to evaluate the exponents separately:

$\left(\frac{a}{b}\right)^{-4} = \frac{b^4}{a^4}$

$\left(\frac{a}{b}\right)^{-2} = \frac{b^2}{a^2}$

Then, we can multiply the two expressions together:

$\frac{b^4}{a^4} \cdot \frac{b^2}{a^2} = \frac{b^6}{a^6}$

Q: What is the final answer to the original problem?

A: The final answer to the original problem is that Expression A is equal to 1, while Expression B is equal to $\frac{b^4}{a^4}$. Therefore, Lola is incorrect.

Q: Can you provide a summary of the key concepts discussed in this article?

A: Yes, here's a summary of the key concepts discussed in this article:

• The correct order of operations is Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).
• When an expression is raised to the power of zero, the result is always 1, unless the expression is equal to zero.
• To evaluate an expression with multiple exponents, you need to follow the order of operations and evaluate each exponent separately.
• To compare the values of two expressions, you need to evaluate each expression separately and then compare their results.

Conclusion

In conclusion, we have discussed the importance of following the correct order of operations when evaluating mathematical expressions. We have also provided examples of how to evaluate expressions with multiple exponents and how to compare the values of two expressions. By following the correct order of operations and carefully evaluating each step, we can ensure accurate results and avoid errors.