Evaluate The Expression: ( 25 C − 1 5 ) 2 \left(\frac{25 C^{-1}}{5}\right)^2 ( 5 25 C − 1 ​ ) 2

Introduction

In mathematics, expressions involving exponents and fractions can be complex and challenging to evaluate. The given expression, $\left(\frac{25 c^{-1}}{5}\right)^2$, requires careful manipulation of exponents and fractions to simplify it. In this article, we will evaluate the expression step by step, using the rules of exponents and fraction arithmetic.

Understanding the Expression

The given expression is $\left(\frac{25 c^{-1}}{5}\right)^2$. To evaluate this expression, we need to understand the rules of exponents and fraction arithmetic. The expression involves a fraction with a negative exponent, which can be challenging to handle.

Simplifying the Expression

To simplify the expression, we can start by evaluating the fraction inside the parentheses. We can rewrite the fraction as $\frac{25}{5c}$. This simplifies the expression to $\left(\frac{5}{c}\right)^2$.

Applying the Power Rule

The power rule states that for any non-zero number $a$ and integers $m$ and $n$, $(a^m)^n = a^{mn}$. We can apply this rule to the expression $\left(\frac{5}{c}\right)^2$. This simplifies the expression to $\frac{5^2}{c^2}$.

Evaluating the Exponents

The expression $\frac{5^2}{c^2}$ involves exponents that need to be evaluated. We can rewrite $5^2$ as $25$ and $c^2$ as $c^2$. This simplifies the expression to $\frac{25}{c^2}$.

Simplifying the Fraction

The expression $\frac{25}{c^2}$ involves a fraction that can be simplified. We can rewrite the fraction as $\frac{25}{c^2} = \frac{25}{c^2} \cdot \frac{c^2}{c^2}$. This simplifies the expression to $\frac{25c^2}{c^4}$.

Canceling Out the Common Factors

The expression $\frac{25c^2}{c^4}$ involves common factors that can be canceled out. We can rewrite the expression as $\frac{25c^2}{c^4} = \frac{25}{c^2}$. This simplifies the expression to $\frac{25}{c^2}$.

Final Answer

The final answer to the expression $\left(\frac{25 c^{-1}}{5}\right)^2$ is $\frac{25}{c^2}$.

Conclusion

Evaluating the expression $\left(\frac{25 c^{-1}}{5}\right)^2$ requires careful manipulation of exponents and fractions. By applying the rules of exponents and fraction arithmetic, we can simplify the expression to $\frac{25}{c^2}$. This example demonstrates the importance of understanding the rules of exponents and fraction arithmetic in evaluating complex mathematical expressions.

Frequently Asked Questions

• Q: What is the rule for evaluating exponents? A: The rule for evaluating exponents is $(a^m)^n = a^{mn}$.
• Q: How do you simplify a fraction with a negative exponent? A: To simplify a fraction with a negative exponent, you can rewrite the fraction as $\frac{1}{a^{-n}}$.
• Q: What is the final answer to the expression $\left(\frac{25 c^{-1}}{5}\right)^2$? A: The final answer to the expression $\left(\frac{25 c^{-1}}{5}\right)^2$ is $\frac{25}{c^2}$.

Further Reading

• Exponents and Fractions: A Comprehensive Guide
• Evaluating Exponents: Rules and Examples
• Simplifying Fractions: A Step-by-Step Guide
  

Introduction

Evaluating expressions with exponents and fractions can be challenging, but with the right rules and techniques, it can be done with ease. In this article, we will answer some frequently asked questions about evaluating expressions with exponents and fractions.

Q&A

Q: What is the rule for evaluating exponents?

A: The rule for evaluating exponents is $(a^m)^n = a^{mn}$. This means that when you have an exponent raised to another exponent, you can multiply the exponents together.

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

A: To simplify a fraction with a negative exponent, you can rewrite the fraction as $\frac{1}{a^{-n}}$. This is because $a^{-n} = \frac{1}{a^n}$.

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

A: A positive exponent means that the base is raised to a power, while a negative exponent means that the base is raised to a power and then taken as a reciprocal.

Q: How do you evaluate an expression with multiple exponents?

A: To evaluate an expression with multiple exponents, you can follow the order of operations (PEMDAS):

1. Evaluate the expressions inside the parentheses.
2. Evaluate any exponents.
3. Multiply or divide from left to right.

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

A: The rule for multiplying fractions with exponents is $\frac{a^m}{b^n} \cdot \frac{c^p}{d^q} = \frac{a^mc^p}{b^nd^q}$. This means that when you multiply fractions with exponents, you can multiply the numerators and denominators separately.

Q: How do you simplify an expression with a negative exponent in the denominator?

A: To simplify an expression with a negative exponent in the denominator, you can rewrite the expression as $\frac{a^m}{a^{-n}} = a^{m+n}$. This is because $a^{-n} = \frac{1}{a^n}$.

Q: What is the final answer to the expression $\left(\frac{25 c^{-1}}{5}\right)^2$?

A: The final answer to the expression $\left(\frac{25 c^{-1}}{5}\right)^2$ is $\frac{25}{c^2}$.

Conclusion

Evaluating expressions with exponents and fractions can be challenging, but with the right rules and techniques, it can be done with ease. By following the order of operations and applying the rules for exponents and fractions, you can simplify complex expressions and arrive at the correct answer.

Frequently Asked Questions

• Q: What is the rule for evaluating exponents? A: The rule for evaluating exponents is $(a^m)^n = a^{mn}$.
• Q: How do you simplify a fraction with a negative exponent? A: To simplify a fraction with a negative exponent, you can rewrite the fraction as $\frac{1}{a^{-n}}$.
• Q: What is the difference between a positive and negative exponent? A: A positive exponent means that the base is raised to a power, while a negative exponent means that the base is raised to a power and then taken as a reciprocal.

Further Reading

• Exponents and Fractions: A Comprehensive Guide
• Evaluating Exponents: Rules and Examples
• Simplifying Fractions: A Step-by-Step Guide

Additional Resources

• Khan Academy: Exponents and Fractions
• Mathway: Exponents and Fractions Calculator
• Wolfram Alpha: Exponents and Fractions Calculator