Gerardo Is Simplifying Expressions With Very Large Exponents. He Arrives At Each Of The Results Below. For Each Result, Decide If He Is Correct And Justify Your Answer Using The Meaning Of Exponents.a. $ \frac{x {150}}{x {50}} \Rightarrow X^3

Understanding Exponents and Their Meaning

Exponents are a fundamental concept in mathematics, representing the number of times a base number is multiplied by itself. For instance, in the expression $x^3$, the base is $x$ and the exponent is $3$, indicating that $x$ is multiplied by itself three times. This results in $x \times x \times x = x^3$. Exponents can be added, subtracted, multiplied, and divided, following specific rules that govern their behavior.

Gerardo's Results: A Critical Analysis

Gerardo is working with expressions that involve very large exponents. We will examine each of his results, applying the rules of exponents to determine if he is correct or not.

Result 1: $ \frac{x^{150}}{x{50}} \Rightarrow x^3$

Is Gerardo Correct?

To determine if Gerardo is correct, we need to apply the rule of dividing exponents with the same base. When dividing exponents with the same base, we subtract the exponents. In this case, we have:

$$\frac{x^{150}}{x^{50}} = x^{150-50} = x^{100}$$

Not $x^3$ as Gerardo claims. Therefore, Gerardo is incorrect in this case.

Why is Gerardo's Answer Incorrect?

Gerardo's answer is incorrect because he failed to apply the correct rule for dividing exponents with the same base. When dividing exponents with the same base, we subtract the exponents, not multiply them. This fundamental mistake led to an incorrect result.

Result 2: $ \frac{x^{200}}{x{100}} \Rightarrow x^{100}$

Is Gerardo Correct?

To determine if Gerardo is correct, we need to apply the rule of dividing exponents with the same base. When dividing exponents with the same base, we subtract the exponents. In this case, we have:

$$\frac{x^{200}}{x^{100}} = x^{200-100} = x^{100}$$

Gerardo is correct in this case.

Why is Gerardo's Answer Correct?

Gerardo's answer is correct because he applied the correct rule for dividing exponents with the same base. When dividing exponents with the same base, we subtract the exponents, which in this case results in $x^{100}$.

Result 3: $ \frac{x^{300}}{x{200}} \Rightarrow x^{100}$

Is Gerardo Correct?

To determine if Gerardo is correct, we need to apply the rule of dividing exponents with the same base. When dividing exponents with the same base, we subtract the exponents. In this case, we have:

$$\frac{x^{300}}{x^{200}} = x^{300-200} = x^{100}$$

Gerardo is correct in this case.

Why is Gerardo's Answer Correct?

Gerardo's answer is correct because he applied the correct rule for dividing exponents with the same base. When dividing exponents with the same base, we subtract the exponents, which in this case results in $x^{100}$.

Result 4: $ \frac{x^{400}}{x{300}} \Rightarrow x^{100}$

Is Gerardo Correct?

To determine if Gerardo is correct, we need to apply the rule of dividing exponents with the same base. When dividing exponents with the same base, we subtract the exponents. In this case, we have:

$$\frac{x^{400}}{x^{300}} = x^{400-300} = x^{100}$$

Gerardo is correct in this case.

Why is Gerardo's Answer Correct?

Gerardo's answer is correct because he applied the correct rule for dividing exponents with the same base. When dividing exponents with the same base, we subtract the exponents, which in this case results in $x^{100}$.

Conclusion

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In conclusion, Gerardo's results are a mix of correct and incorrect answers. He correctly applied the rule of dividing exponents with the same base in cases 2, 3, and 4, resulting in $x^{100}$. However, he incorrectly applied the rule in case 1, resulting in $x^3$. This highlights the importance of understanding and applying the rules of exponents correctly to achieve accurate results.

Tips for Simplifying Expressions with Large Exponents

When simplifying expressions with large exponents, remember to:

• Apply the correct rule for dividing exponents with the same base (subtract the exponents)
• Use the correct order of operations (PEMDAS/BODMAS)
• Simplify the expression step by step, avoiding unnecessary calculations
• Double-check your work to ensure accuracy

Q: What is the rule for dividing exponents with the same base?

A: When dividing exponents with the same base, we subtract the exponents. For example, $\frac{x^a}{x^b} = x^{a-b}$.

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

A: To simplify an expression with a large exponent, follow these steps:

1. Identify the base and exponent.
2. Apply the correct rule for dividing exponents with the same base (subtract the exponents).
3. Simplify the expression step by step, avoiding unnecessary calculations.
4. Double-check your work to ensure accuracy.

Q: What is the difference between $x^a$ and $x^{a-b}$?

A: $x^a$ represents the base $x$ multiplied by itself $a$ times, while $x^{a-b}$ represents the base $x$ multiplied by itself $(a-b)$ times.

Q: Can I simplify an expression with a negative exponent?

A: Yes, you can simplify an expression with a negative exponent by applying the rule for negative exponents. For example, $\frac{1}{x^a} = x^{-a}$.

Q: How do I handle expressions with multiple exponents?

A: To handle expressions with multiple exponents, follow these steps:

1. Identify the base and exponents.
2. Apply the correct rule for multiplying exponents with the same base (add the exponents).
3. Simplify the expression step by step, avoiding unnecessary calculations.
4. Double-check your work to ensure accuracy.

Q: What is the rule for multiplying exponents with the same base?

A: When multiplying exponents with the same base, we add the exponents. For example, $x^a \times x^b = x^{a+b}$.

Q: Can I simplify an expression with a zero exponent?

A: Yes, you can simplify an expression with a zero exponent by applying the rule for zero exponents. For example, $x^0 = 1$.

Q: How do I handle expressions with fractional exponents?

A: To handle expressions with fractional exponents, follow these steps:

1. Identify the base and exponents.
2. Apply the correct rule for fractional exponents (take the square root of the base raised to the power of the exponent).
3. Simplify the expression step by step, avoiding unnecessary calculations.
4. Double-check your work to ensure accuracy.

Q: What is the rule for simplifying expressions with large exponents?

A: To simplify expressions with large exponents, follow these steps:

1. Identify the base and exponent.
2. Apply the correct rule for dividing exponents with the same base (subtract the exponents).
3. Simplify the expression step by step, avoiding unnecessary calculations.
4. Double-check your work to ensure accuracy.

Q: Can I use a calculator to simplify expressions with large exponents?

A: Yes, you can use a calculator to simplify expressions with large exponents. However, be sure to double-check your work to ensure accuracy.

Q: How do I know if I have simplified an expression correctly?

A: To ensure that you have simplified an expression correctly, follow these steps:

1. Check your work for accuracy.
2. Verify that you have applied the correct rules for exponents.
3. Simplify the expression step by step, avoiding unnecessary calculations.
4. Double-check your work to ensure accuracy.

By following these tips and answering these frequently asked questions, you can become more confident in your ability to simplify expressions with large exponents.