Simplify ( 2 4 ) − 1 \left(2^4\right)^{-1} ( 2 4 ) − 1 .A. 1 2 − 4 \frac{1}{2^{-4}} 2 − 4 1 ​ B. 1 2 4 \frac{1}{2^4} 2 4 1 ​ C. − 2 4 -2^4 − 2 4 D. 2 3 2^3 2 3

Introduction

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 $\left(2^4\right)^{-1}$ using the properties of exponents.

Understanding Exponents

Exponents are a shorthand way of representing repeated multiplication. For example, $2^4$ means $2$ multiplied by itself $4$ times, which is equal to $16$. When we have an exponent raised to another exponent, we can simplify it using the property of exponents that states $(a^m)^n = a^{mn}$.

Simplifying the Expression

Let's start by simplifying the expression $\left(2^4\right)^{-1}$. Using the property of exponents mentioned above, we can rewrite the expression as $2^{4(-1)}$. Now, we need to simplify the exponent $4(-1)$.

Multiplying Exponents

When we multiply two exponents with the same base, we can add their exponents. In this case, we have $4(-1)$, which is equal to $-4$. Therefore, the expression becomes $2^{-4}$.

Negative Exponents

A negative exponent is equal to the reciprocal of the positive exponent. In other words, $a^{-n} = \frac{1}{a^n}$. Therefore, we can rewrite the expression $2^{-4}$ as $\frac{1}{2^4}$.

Simplifying the Final Expression

Now that we have simplified the expression $\left(2^4\right)^{-1}$ to $\frac{1}{2^4}$, we can see that it matches option B. Therefore, the correct answer is $\frac{1}{2^4}$.

Conclusion

In this article, we have learned how to simplify the expression $\left(2^4\right)^{-1}$ using the properties of exponents. We have seen how to multiply exponents and how to handle negative exponents. By following these steps, we can simplify complex expressions and arrive at the correct answer.

Common Mistakes to Avoid

When simplifying expressions with exponents, it's essential to remember the following common mistakes to avoid:

• Not following the order of operations: When simplifying expressions, it's crucial to follow the order of operations (PEMDAS): Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction.
• Not using the correct property of exponents: When simplifying expressions with exponents, it's essential to use the correct property of exponents, such as $(a^m)^n = a^{mn}$.
• Not handling negative exponents correctly: Negative exponents can be tricky, but it's essential to remember that $a^{-n} = \frac{1}{a^n}$.

Practice Problems

To reinforce your understanding of simplifying expressions with exponents, try the following practice problems:

• Simplify the expression $\left(3^2\right)^{-3}$.
• Simplify the expression $\left(4^{-2}\right)^{-1}$.
• Simplify the expression $\left(2^3\right)^{-2}$.

Answer Key

Here are the answers to the practice problems:

• $\left(3^2\right)^{-3} = \frac{1}{3^6}$
• $\left(4^{-2}\right)^{-1} = 4^2$
• $\left(2^3\right)^{-2} = \frac{1}{2^6}$

Conclusion

Introduction

In our previous article, we explored the concept of simplifying exponents and provided a step-by-step guide on how to simplify the expression $\left(2^4\right)^{-1}$. In this article, we will answer some frequently asked questions about simplifying exponents to help you better understand this concept.

Q&A

Q: What is the order of operations when simplifying expressions with exponents?

A: When simplifying expressions with exponents, it's essential to follow the order of operations (PEMDAS):

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

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

A: A negative exponent is equal to the reciprocal of the positive exponent. In other words, $a^{-n} = \frac{1}{a^n}$. For example, $2^{-3} = \frac{1}{2^3}$.

Q: Can I simplify an expression with a negative exponent by multiplying it by a positive exponent?

A: Yes, you can simplify an expression with a negative exponent by multiplying it by a positive exponent. For example, $2^{-3} \cdot 2^3 = 2^{-3+3} = 2^0 = 1$.

Q: How do I simplify an expression with multiple exponents?

A: When simplifying an expression with multiple exponents, you can use the property of exponents that states $(a^m)^n = a^{mn}$. For example, $(2^3)^4 = 2^{3 \cdot 4} = 2^{12}$.

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

A: Yes, you can simplify an expression with a zero exponent. Any number raised to the power of zero is equal to 1. For example, $2^0 = 1$.

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

A: A fractional exponent is equal to the square root of the number raised to the power of the numerator, divided by the square root of the number raised to the power of the denominator. For example, $2^{\frac{1}{2}} = \sqrt{2}$.

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

A: Yes, you can simplify an expression with a negative fractional exponent. A negative fractional exponent is equal to the reciprocal of the positive fractional exponent. For example, $2^{-\frac{1}{2}} = \frac{1}{\sqrt{2}}$.

Conclusion

In conclusion, simplifying exponents requires a solid understanding of the properties of exponents and how to handle negative exponents, multiple exponents, zero exponents, and fractional exponents. By following the steps outlined in this article and practicing with the provided examples, you can become proficient in simplifying complex expressions and arrive at the correct answer.

Practice Problems

To reinforce your understanding of simplifying expressions with exponents, try the following practice problems:

• Simplify the expression $\left(3^2\right)^{-3}$.
• Simplify the expression $\left(4^{-2}\right)^{-1}$.
• Simplify the expression $\left(2^3\right)^{-2}$.
• Simplify the expression $2^{-3} \cdot 2^3$.
• Simplify the expression $(2^3)^4$.
• Simplify the expression $2^0$.
• Simplify the expression $2^{\frac{1}{2}}$.
• Simplify the expression $2^{-\frac{1}{2}}$.

Answer Key

Here are the answers to the practice problems:

• $\left(3^2\right)^{-3} = \frac{1}{3^6}$
• $\left(4^{-2}\right)^{-1} = 4^2$
• $\left(2^3\right)^{-2} = \frac{1}{2^6}$
• $2^{-3} \cdot 2^3 = 2^{-3+3} = 2^0 = 1$
• $(2^3)^4 = 2^{3 \cdot 4} = 2^{12}$
• $2^0 = 1$
• $2^{\frac{1}{2}} = \sqrt{2}$
• $2^{-\frac{1}{2}} = \frac{1}{\sqrt{2}}$