Simplify. Express Your Answer As A Single Term Using Exponents. 964 42 964 − 1 \frac{964^{42}}{964^{-1}} 96 4 − 1 96 4 42 ​

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Understanding Exponents and Their Rules

When dealing with exponents, it's essential to understand the rules that govern their behavior. Exponents are a shorthand way of representing repeated multiplication of a number. For example, $a^m$ represents the product $a \cdot a \cdot a \cdot \ldots \cdot a$ (m times). The exponentiation operation is commutative, meaning that the order of the factors does not change the result. However, when dealing with fractions that involve exponents, we need to apply the rules of exponentiation carefully.

Applying the Quotient Rule for Exponents

The quotient rule for exponents states that when dividing two powers with the same base, we subtract the exponents. Mathematically, this can be represented as:

$$\frac{a^m}{a^n} = a^{m-n}$$

This rule is crucial in simplifying expressions that involve exponents. In the given problem, we have the expression $\frac{964^{42}}{964^{-1}}$. To simplify this expression, we can apply the quotient rule for exponents.

Simplifying the Expression

Using the quotient rule for exponents, we can rewrite the given expression as:

$$\frac{964^{42}}{964^{-1}} = 964^{42-(-1)}$$

Now, we can simplify the exponent by subtracting the two numbers:

$$964^{42-(-1)} = 964^{43}$$

Understanding the Concept of Negative Exponents

In the previous step, we encountered a negative exponent. Negative exponents can be rewritten as positive exponents by taking the reciprocal of the base. Mathematically, this can be represented as:

$$a^{-m} = \frac{1}{a^m}$$

This concept is essential in simplifying expressions that involve negative exponents.

Rewriting the Expression with a Positive Exponent

Using the concept of negative exponents, we can rewrite the expression $964^{43}$ as:

$$964^{43} = \frac{964^{43}}{1}$$

Now, we can rewrite the expression with a positive exponent by taking the reciprocal of the base:

$$\frac{964^{43}}{1} = \frac{1}{(964^{-1})^{43}}$$

Simplifying the Expression Further

Using the concept of negative exponents, we can rewrite the expression $(964^{-1})^{43}$ as:

$$(964^{-1})^{43} = 964^{-43}$$

Now, we can simplify the expression by applying the quotient rule for exponents:

$$\frac{1}{(964^{-1})^{43}} = 964^{43}$$

Conclusion

In conclusion, the expression $\frac{964^{42}}{964^{-1}}$ can be simplified using the quotient rule for exponents and the concept of negative exponents. By applying these rules, we can rewrite the expression as a single term using exponents.

Final Answer

The final answer is: $\boxed{964^{43}}$

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Frequently Asked Questions

Q: What is the quotient rule for exponents?

A: The quotient rule for exponents states that when dividing two powers with the same base, we subtract the exponents. Mathematically, this can be represented as:

$$\frac{a^m}{a^n} = a^{m-n}$$

Q: How do I apply the quotient rule for exponents?

A: To apply the quotient rule for exponents, simply subtract the exponents of the two powers with the same base. For example, if we have the expression $\frac{a^m}{a^n}$, we can rewrite it as $a^{m-n}$.

Q: What is the concept of negative exponents?

A: Negative exponents can be rewritten as positive exponents by taking the reciprocal of the base. Mathematically, this can be represented as:

$$a^{-m} = \frac{1}{a^m}$$

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

A: To rewrite an expression with a negative exponent, simply take the reciprocal of the base. For example, if we have the expression $a^{-m}$, we can rewrite it as $\frac{1}{a^m}$.

Q: Can I simplify an expression with a negative exponent using the quotient rule for exponents?

A: Yes, you can simplify an expression with a negative exponent using the quotient rule for exponents. For example, if we have the expression $\frac{a^m}{a^{-n}}$, we can rewrite it as $a^{m+n}$.

Q: What is the final answer to the expression $\frac{964^{42}}{964^{-1}}$?

A: The final answer to the expression $\frac{964^{42}}{964^{-1}}$ is $964^{43}$.

Q: Why is it important to understand the rules of exponents?

A: Understanding the rules of exponents is crucial in simplifying expressions that involve exponents. By applying the rules of exponents, you can rewrite complex expressions into simpler ones, making it easier to solve problems.

Q: Can I use the quotient rule for exponents to simplify expressions with different bases?

A: No, the quotient rule for exponents only applies to expressions with the same base. If you have an expression with different bases, you cannot use the quotient rule for exponents to simplify it.

Q: How do I know when to use the quotient rule for exponents?

A: You should use the quotient rule for exponents when you have an expression with the same base and you want to simplify it. The quotient rule for exponents is a powerful tool that can help you simplify complex expressions and make them easier to solve.

Additional Resources

• Exponents and Exponent Rules
• Quotient Rule for Exponents
• Negative Exponents

Conclusion

In conclusion, the expression $\frac{964^{42}}{964^{-1}}$ can be simplified using the quotient rule for exponents and the concept of negative exponents. By applying these rules, we can rewrite the expression as a single term using exponents. We hope this Q&A article has helped you understand the rules of exponents and how to apply them to simplify complex expressions.