Evaluate The Expression:$\left(25^{-\frac{3}{2}}\right)^{\frac{1}{3}} = \square$

by ADMIN 81 views

Introduction

In mathematics, exponential expressions are a fundamental concept that plays a crucial role in various mathematical operations. Evaluating these expressions requires a deep understanding of the underlying mathematical principles and rules. In this article, we will focus on evaluating the expression (25βˆ’32)13\left(25^{-\frac{3}{2}}\right)^{\frac{1}{3}} and provide a step-by-step guide on how to simplify it.

Understanding Exponents

Before we dive into the evaluation process, it's essential to understand the concept of exponents. An exponent is a small number that is written above and to the right of a number, indicating how many times the base number should be multiplied by itself. For example, in the expression 232^3, the exponent 33 indicates that the base number 22 should be multiplied by itself three times, resulting in 2Γ—2Γ—2=82 \times 2 \times 2 = 8.

Evaluating the Expression

Now that we have a basic understanding of exponents, let's focus on evaluating the expression (25βˆ’32)13\left(25^{-\frac{3}{2}}\right)^{\frac{1}{3}}. To simplify this expression, we need to follow the order of operations (PEMDAS):

  1. Parentheses: Evaluate the expression inside the parentheses first.
  2. Exponents: Evaluate any exponents 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.

Step 1: Evaluate the Expression Inside the Parentheses

The expression inside the parentheses is 25βˆ’3225^{-\frac{3}{2}}. To evaluate this expression, we need to follow the rules of exponents. When a negative exponent is raised to a power, we can rewrite it as a positive exponent by taking the reciprocal of the base and changing the sign of the exponent.

25βˆ’32=1253225^{-\frac{3}{2}} = \frac{1}{25^{\frac{3}{2}}}

Step 2: Simplify the Expression

Now that we have simplified the expression inside the parentheses, let's focus on simplifying the entire expression. We can start by rewriting the expression as:

(12532)13\left(\frac{1}{25^{\frac{3}{2}}}\right)^{\frac{1}{3}}

Step 3: Apply the Power Rule

The power rule states that when a power is raised to a power, we can multiply the exponents. In this case, we have:

(12532)13=1(2532)13\left(\frac{1}{25^{\frac{3}{2}}}\right)^{\frac{1}{3}} = \frac{1}{\left(25^{\frac{3}{2}}\right)^{\frac{1}{3}}}

Step 4: Simplify the Expression

Now that we have applied the power rule, let's simplify the expression further. We can start by rewriting the expression as:

12512\frac{1}{25^{\frac{1}{2}}}

Step 5: Evaluate the Final Expression

The final expression is 12512\frac{1}{25^{\frac{1}{2}}}. To evaluate this expression, we need to find the square root of 2525 and take the reciprocal.

12512=15\frac{1}{25^{\frac{1}{2}}} = \frac{1}{5}

Conclusion

In this article, we evaluated the expression (25βˆ’32)13\left(25^{-\frac{3}{2}}\right)^{\frac{1}{3}} and provided a step-by-step guide on how to simplify it. We followed the order of operations (PEMDAS) and applied the rules of exponents to simplify the expression. The final answer is 15\frac{1}{5}.

Common Mistakes to Avoid

When evaluating exponential expressions, it's essential to follow the order of operations (PEMDAS) and apply the rules of exponents correctly. Some common mistakes to avoid include:

  • Not following the order of operations (PEMDAS)
  • Not applying the rules of exponents correctly
  • Not simplifying the expression correctly

Real-World Applications

Exponential expressions have numerous real-world applications in various fields, including:

  • Science: Exponential expressions are used to model population growth, chemical reactions, and other scientific phenomena.
  • Engineering: Exponential expressions are used to design and optimize systems, such as electrical circuits and mechanical systems.
  • Finance: Exponential expressions are used to calculate interest rates and investment returns.

Final Thoughts

Introduction

In our previous article, we evaluated the expression (25βˆ’32)13\left(25^{-\frac{3}{2}}\right)^{\frac{1}{3}} and provided a step-by-step guide on how to simplify it. In this article, we will answer some frequently asked questions (FAQs) related to evaluating exponential expressions.

Q&A

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that dictates the order in which mathematical operations should be performed. The acronym PEMDAS stands for:

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

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

A: When a negative exponent is raised to a power, we can rewrite it as a positive exponent by taking the reciprocal of the base and changing the sign of the exponent. For example:

25βˆ’32=1253225^{-\frac{3}{2}} = \frac{1}{25^{\frac{3}{2}}}

Q: How do I apply the power rule?

A: The power rule states that when a power is raised to a power, we can multiply the exponents. For example:

(12532)13=1(2532)13\left(\frac{1}{25^{\frac{3}{2}}}\right)^{\frac{1}{3}} = \frac{1}{\left(25^{\frac{3}{2}}\right)^{\frac{1}{3}}}

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

A: A power is a number that is raised to a certain power, while an exponent is the number that is raised to a certain power. For example:

  • 232^3 is a power, where 2 is the base and 3 is the exponent.
  • 323^2 is also a power, where 3 is the base and 2 is the exponent.

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

A: To simplify an expression with multiple exponents, we need to follow the order of operations (PEMDAS) and apply the rules of exponents correctly. For example:

(25βˆ’32)13=12512\left(25^{-\frac{3}{2}}\right)^{\frac{1}{3}} = \frac{1}{25^{\frac{1}{2}}}

Q: What are some common mistakes to avoid when evaluating exponential expressions?

A: Some common mistakes to avoid when evaluating exponential expressions include:

  • Not following the order of operations (PEMDAS)
  • Not applying the rules of exponents correctly
  • Not simplifying the expression correctly

Q: What are some real-world applications of exponential expressions?

A: Exponential expressions have numerous real-world applications in various fields, including:

  • Science: Exponential expressions are used to model population growth, chemical reactions, and other scientific phenomena.
  • Engineering: Exponential expressions are used to design and optimize systems, such as electrical circuits and mechanical systems.
  • Finance: Exponential expressions are used to calculate interest rates and investment returns.

Conclusion

Evaluating exponential expressions requires a deep understanding of the underlying mathematical principles and rules. By following the order of operations (PEMDAS) and applying the rules of exponents correctly, we can simplify complex expressions and arrive at the correct solution. In this article, we answered some frequently asked questions (FAQs) related to evaluating exponential expressions. We hope this article has provided you with a better understanding of exponential expressions and how to evaluate them correctly.

Additional Resources

For more information on evaluating exponential expressions, we recommend the following resources:

  • Mathway: A math problem solver that can help you evaluate exponential expressions.
  • Khan Academy: A free online resource that provides video lessons and practice exercises on evaluating exponential expressions.
  • Wolfram Alpha: A computational knowledge engine that can help you evaluate exponential expressions and provide step-by-step solutions.

Final Thoughts

Evaluating exponential expressions is a crucial skill that is used in various fields, including science, engineering, and finance. By following the order of operations (PEMDAS) and applying the rules of exponents correctly, we can simplify complex expressions and arrive at the correct solution. We hope this article has provided you with a better understanding of exponential expressions and how to evaluate them correctly.