Which Of The Following Is Equivalent To $\frac{1}{2^{-3}}$?A. $-8$ B. $-\frac{1}{8}$ C. $\frac{1}{8}$ D. $8$

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Understanding Exponents and Fractions

When dealing with exponents and fractions, it's essential to understand the rules and properties that govern their behavior. In this article, we'll focus on simplifying expressions involving exponents and fractions, specifically the expression 12−3\frac{1}{2^{-3}}. We'll explore the different options provided and determine which one is equivalent to the given expression.

The Power of a Power Rule

The power of a power rule states that when we raise a power to another power, we multiply the exponents. In other words, (am)n=amn(a^m)^n = a^{mn}. This rule can be applied to simplify expressions involving exponents.

Simplifying the Expression

Let's start by simplifying the expression 12−3\frac{1}{2^{-3}}. To do this, we can apply the power of a power rule. We can rewrite 2−32^{-3} as (23)−1(2^3)^{-1}, which is equivalent to 123\frac{1}{2^3}.

Applying the Power of a Power Rule

Using the power of a power rule, we can simplify the expression further. We have 123=18\frac{1}{2^3} = \frac{1}{8}.

Evaluating the Options

Now that we've simplified the expression, let's evaluate the options provided:

A. −8-8 B. −18-\frac{1}{8} C. 18\frac{1}{8} D. 88

Which Option is Equivalent?

Based on our simplification, we can see that option C, 18\frac{1}{8}, is equivalent to the expression 12−3\frac{1}{2^{-3}}.

Conclusion

In conclusion, when dealing with exponents and fractions, it's essential to understand the rules and properties that govern their behavior. By applying the power of a power rule, we can simplify expressions involving exponents and fractions. In this article, we've explored the expression 12−3\frac{1}{2^{-3}} and determined that option C, 18\frac{1}{8}, is equivalent to the given expression.

Common Mistakes to Avoid

When simplifying expressions involving exponents and fractions, it's essential to avoid common mistakes. Here are a few common mistakes to watch out for:

  • Not applying the power of a power rule: Failing to apply the power of a power rule can lead to incorrect simplifications.
  • Not following the order of operations: Failing to follow the order of operations (PEMDAS) can lead to incorrect simplifications.
  • Not simplifying fractions: Failing to simplify fractions can lead to incorrect simplifications.

Tips and Tricks

Here are a few tips and tricks to help you simplify expressions involving exponents and fractions:

  • Use the power of a power rule: The power of a power rule is a powerful tool for simplifying expressions involving exponents.
  • Follow the order of operations: Following the order of operations (PEMDAS) is essential for simplifying expressions involving exponents and fractions.
  • Simplify fractions: Simplifying fractions is essential for simplifying expressions involving exponents and fractions.

Real-World Applications

Simplifying expressions involving exponents and fractions has numerous real-world applications. Here are a few examples:

  • Science and Engineering: Simplifying expressions involving exponents and fractions is essential in science and engineering, where complex calculations are often required.
  • Finance: Simplifying expressions involving exponents and fractions is essential in finance, where complex financial calculations are often required.
  • Computer Science: Simplifying expressions involving exponents and fractions is essential in computer science, where complex algorithms are often required.

Conclusion

In conclusion, simplifying expressions involving exponents and fractions is a crucial skill in mathematics. By understanding the rules and properties that govern their behavior, we can simplify complex expressions and arrive at accurate solutions. In this article, we've explored the expression 12−3\frac{1}{2^{-3}} and determined that option C, 18\frac{1}{8}, is equivalent to the given expression.

Frequently Asked Questions

In this article, we'll address some of the most frequently asked questions about simplifying exponents and fractions.

Q: What is the power of a power rule?

A: The power of a power rule states that when we raise a power to another power, we multiply the exponents. In other words, (am)n=amn(a^m)^n = a^{mn}.

Q: How do I simplify an expression involving exponents and fractions?

A: To simplify an expression involving exponents and fractions, you can follow these steps:

  1. Apply the power of a power rule to simplify the exponents.
  2. Simplify the fractions by dividing the numerator and denominator by their greatest common divisor.
  3. Combine the simplified exponents and fractions to arrive at the final answer.

Q: What is the order of operations?

A: The order of operations is a set of rules that dictate the order in which we perform mathematical operations. The order of operations is:

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

Q: How do I simplify a fraction with a negative exponent?

A: To simplify a fraction with a negative exponent, you can follow these steps:

  1. Rewrite the fraction with a positive exponent by taking the reciprocal of the fraction.
  2. Simplify the fraction by dividing the numerator and denominator by their greatest common divisor.
  3. Combine the simplified fraction to arrive at the final answer.

Q: What is the difference between a positive and negative exponent?

A: A positive exponent indicates that the base is raised to a power, while a negative exponent indicates that the reciprocal of the base is raised to a power.

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

A: To simplify an expression involving multiple exponents, you can follow these steps:

  1. Apply the power of a power rule to simplify the exponents.
  2. Simplify the fractions by dividing the numerator and denominator by their greatest common divisor.
  3. Combine the simplified exponents and fractions to arrive at the final answer.

Q: What are some common mistakes to avoid when simplifying exponents and fractions?

A: Some common mistakes to avoid when simplifying exponents and fractions include:

  • Not applying the power of a power rule
  • Not following the order of operations
  • Not simplifying fractions
  • Not combining simplified exponents and fractions

Q: How do I apply the power of a power rule to simplify an expression?

A: To apply the power of a power rule, you can follow these steps:

  1. Identify the exponent that is being raised to another power.
  2. Multiply the exponents by applying the power of a power rule.
  3. Simplify the resulting expression.

Q: What are some real-world applications of simplifying exponents and fractions?

A: Some real-world applications of simplifying exponents and fractions include:

  • Science and engineering
  • Finance
  • Computer science
  • Data analysis

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

A: To simplify an expression involving a variable with an exponent, you can follow these steps:

  1. Identify the variable and its exponent.
  2. Apply the power of a power rule to simplify the exponent.
  3. Simplify the resulting expression.

Q: What is the difference between a variable with a positive and negative exponent?

A: A variable with a positive exponent indicates that the variable is raised to a power, while a variable with a negative exponent indicates that the reciprocal of the variable is raised to a power.

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

A: To simplify an expression involving multiple variables with exponents, you can follow these steps:

  1. Identify the variables and their exponents.
  2. Apply the power of a power rule to simplify the exponents.
  3. Simplify the resulting expression.

Q: What are some tips and tricks for simplifying exponents and fractions?

A: Some tips and tricks for simplifying exponents and fractions include:

  • Using the power of a power rule
  • Following the order of operations
  • Simplifying fractions
  • Combining simplified exponents and fractions

Q: How do I check my work when simplifying exponents and fractions?

A: To check your work when simplifying exponents and fractions, you can follow these steps:

  1. Review the original expression.
  2. Verify that you applied the power of a power rule correctly.
  3. Verify that you simplified the fractions correctly.
  4. Verify that you combined the simplified exponents and fractions correctly.

Q: What are some common errors to watch out for when simplifying exponents and fractions?

A: Some common errors to watch out for when simplifying exponents and fractions include:

  • Not applying the power of a power rule
  • Not following the order of operations
  • Not simplifying fractions
  • Not combining simplified exponents and fractions

Q: How do I use technology to simplify exponents and fractions?

A: To use technology to simplify exponents and fractions, you can follow these steps:

  1. Use a calculator or computer program to simplify the expression.
  2. Verify that the technology applied the power of a power rule correctly.
  3. Verify that the technology simplified the fractions correctly.
  4. Verify that the technology combined the simplified exponents and fractions correctly.

Q: What are some real-world applications of using technology to simplify exponents and fractions?

A: Some real-world applications of using technology to simplify exponents and fractions include:

  • Science and engineering
  • Finance
  • Computer science
  • Data analysis