Which Expression Is Equivalent To $\frac{2^8}{2^{-1}}$?A. $2^8$ B. $2^{-8}$ C. $2^9$ D. $2^7$

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Introduction

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Exponents are a fundamental concept in mathematics, and understanding how to simplify them is crucial for solving various mathematical problems. In this article, we will focus on simplifying the expression $\frac{2^8}{2^{-1}}$ and explore the different options available.

Understanding Exponents

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Exponents are a shorthand way of representing repeated multiplication. For example, $2^3$ can be written as $2 \times 2 \times 2$. When we have a fraction with exponents, we can simplify it by applying the rules of exponents.

Simplifying the Expression

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To simplify the expression $\frac{2^8}{2^{-1}}$, we can use the rule that states $\frac{a^m}{a^n} = a^{m-n}$. In this case, we have:

$$\frac{2^8}{2^{-1}} = 2^{8-(-1)}$$

Evaluating the Exponent

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Now, let's evaluate the exponent $8-(-1)$. When we subtract a negative number, we can rewrite it as adding the positive value. So, $8-(-1)$ is equivalent to $8+1$, which equals $9$.

Simplifying the Expression Further

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Now that we have evaluated the exponent, we can simplify the expression further:

$$2^{8-(-1)} = 2^9$$

Conclusion

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In conclusion, the expression $\frac{2^8}{2^{-1}}$ is equivalent to $2^9$. This is because we can simplify the fraction by applying the rules of exponents, and the resulting expression is $2^9$.

Comparison with Options

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Now, let's compare our result with the options provided:

• A. $2^8$
• B. $2^{-8}$
• C. $2^9$
• D. $2^7$

As we can see, our result matches option C, which is $2^9$.

Final Thoughts

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In this article, we have seen how to simplify the expression $\frac{2^8}{2^{-1}}$ using the rules of exponents. We have also compared our result with the options provided and found that the correct answer is option C, which is $2^9$. This exercise demonstrates the importance of understanding exponents and how to simplify expressions using the rules of exponents.

Frequently Asked Questions

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Q: What is the rule for simplifying fractions with exponents?

A: The rule for simplifying fractions with exponents is $\frac{a^m}{a^n} = a^{m-n}$.

Q: How do I evaluate the exponent $8-(-1)$?

A: To evaluate the exponent $8-(-1)$, we can rewrite it as $8+1$, which equals $9$.

Q: What is the final answer to the expression $\frac{2^8}{2^{-1}}$?

A: The final answer to the expression $\frac{2^8}{2^{-1}}$ is $2^9$.

Additional Resources

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For more information on exponents and how to simplify expressions, we recommend checking out the following resources:

• Khan Academy: Exponents and Exponential Functions
• Mathway: Exponents and Exponential Functions
• Wolfram MathWorld: Exponents and Exponential Functions

By following these resources, you can gain a deeper understanding of exponents and how to simplify expressions using the rules of exponents.

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Introduction

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Exponents and exponential functions are fundamental concepts in mathematics, and understanding them is crucial for solving various mathematical problems. In this article, we will provide a comprehensive Q&A guide to help you better understand exponents and exponential functions.

Q&A

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Q: What is an exponent?

A: An exponent is a small number that is written to the right of a base number and indicates how many times the base number should be multiplied by itself.

Q: What is the difference between an exponent and a power?

A: An exponent is the small number that is written to the right of a base number, while a power is the result of raising the base number to the power of the exponent.

Q: How do I simplify an expression with exponents?

A: To simplify an expression with exponents, you can use the rule that states $\frac{a^m}{a^n} = a^{m-n}$.

Q: What is the rule for multiplying exponents?

A: The rule for multiplying exponents is $a^m \times a^n = a^{m+n}$.

Q: What is the rule for dividing exponents?

A: The rule for dividing exponents is $\frac{a^m}{a^n} = a^{m-n}$.

Q: How do I evaluate an expression with negative exponents?

A: To evaluate an expression with negative exponents, you can rewrite it as a fraction with a positive exponent. For example, $a^{-n}$ is equivalent to $\frac{1}{a^n}$.

Q: What is the difference between an exponential function and a polynomial function?

A: An exponential function is a function that has the form $f(x) = a^x$, where $a$ is a positive constant. A polynomial function, on the other hand, is a function that has the form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.

Q: How do I graph an exponential function?

A: To graph an exponential function, you can use a graphing calculator or a computer program. You can also use a table of values to create a graph.

Q: What is the domain of an exponential function?

A: The domain of an exponential function is all real numbers.

Q: What is the range of an exponential function?

A: The range of an exponential function is all positive real numbers.

Examples

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Example 1: Simplifying an Expression with Exponents

Simplify the expression $\frac{2^8}{2^{-1}}$.

Answer: $2^9$

Example 2: Evaluating an Expression with Negative Exponents

Evaluate the expression $2^{-3}$.

Answer: $\frac{1}{2^3} = \frac{1}{8}$

Example 3: Graphing an Exponential Function

Graph the function $f(x) = 2^x$.

Answer: The graph of the function $f(x) = 2^x$ is an exponential curve that passes through the point $(0, 1)$ and has a horizontal asymptote at $y = 0$.

Conclusion

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In conclusion, exponents and exponential functions are fundamental concepts in mathematics that are used to describe growth and decay. Understanding exponents and exponential functions is crucial for solving various mathematical problems, and this Q&A guide provides a comprehensive overview of the key concepts and rules.

Additional Resources

---

For more information on exponents and exponential functions, we recommend checking out the following resources:

• Khan Academy: Exponents and Exponential Functions
• Mathway: Exponents and Exponential Functions
• Wolfram MathWorld: Exponents and Exponential Functions

By following these resources, you can gain a deeper understanding of exponents and exponential functions and improve your math skills.