Applying the Rule of Negative Exponents

Oct 13, 2025 by miodeya.com 86 views

Introduction

In mathematics, expressions involving exponents and fractions can be complex and challenging to evaluate. The given expression, $ \frac{1}{4^{-2}} $, is a perfect example of such a problem. In this article, we will break down the expression, apply the rules of exponents, and simplify it to its final value.

Understanding Exponents

Before we dive into the expression, let's quickly review the rules of exponents. An exponent is a small number that is placed above and to the right of a number or a variable. It represents the power to which the base is raised. For example, in the expression $ 2^3 $, the exponent 3 represents the power to which the base 2 is raised.

Evaluating the Expression

Now, let's focus on the given expression, $ \frac{1}{4^{-2}} $. To evaluate this expression, we need to apply the rule of negative exponents. A negative exponent is defined as the reciprocal of the positive exponent. In other words, $ a^{-n} = \frac{1}{a^n} $.

Applying the Rule of Negative Exponents

Using the rule of negative exponents, we can rewrite the expression as follows:

$ \frac{1}{4^{-2}} = 4^2 $

Simplifying the Expression

Now that we have applied the rule of negative exponents, we can simplify the expression by evaluating the exponent. In this case, the exponent is 2, which means we need to raise 4 to the power of 2.

Evaluating the Exponent

To evaluate the exponent, we need to multiply 4 by itself 2 times:

$ 4^2 = 4 \times 4 = 16 $

Conclusion

In conclusion, the expression $ \frac{1}{4^{-2}} $ can be evaluated by applying the rule of negative exponents and simplifying the resulting expression. By following the rules of exponents, we can simplify complex expressions and arrive at their final values.

Frequently Asked Questions

What is the rule of negative exponents? The rule of negative exponents states that $ a^{-n} = \frac{1}{a^n} $.
How do I evaluate an expression with a negative exponent? To evaluate an expression with a negative exponent, you need to apply the rule of negative exponents and simplify the resulting expression.
What is the final value of the expression $ \frac{1}{4^{-2}} $? The final value of the expression $ \frac{1}{4^{-2}} $ is 16.

Examples and Applications

Evaluate the expression $ \frac{1}{2^{-3}} $. Using the rule of negative exponents, we can rewrite the expression as follows:

$ \frac{1}{2^{-3}} = 2^3 = 8 $

Evaluate the expression $ \frac{1}{3^{-2}} $. Using the rule of negative exponents, we can rewrite the expression as follows:

$ \frac{1}{3^{-2}} = 3^2 = 9 $

Tips and Tricks

When evaluating expressions with negative exponents, always apply the rule of negative exponents first.
Simplify the resulting expression by evaluating the exponent.
Practice evaluating expressions with negative exponents to become more comfortable with the concept.

Final Thoughts

In conclusion, evaluating expressions with negative exponents requires a clear understanding of the rules of exponents. By applying the rule of negative exponents and simplifying the resulting expression, we can arrive at the final value of the expression. With practice and patience, you can become proficient in evaluating complex expressions and solving mathematical problems with ease.

Introduction

In our previous article, we discussed how to evaluate expressions with negative exponents. We applied the rule of negative exponents and simplified the resulting expression to arrive at the final value. In this article, we will answer some frequently asked questions about evaluating expressions with negative exponents.

Q&A

Q: What is the rule of negative exponents?

A: The rule of negative exponents states that $ a^{-n} = \frac{1}{a^n} $. This means that a negative exponent is defined as the reciprocal of the positive exponent.

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

A: To evaluate an expression with a negative exponent, you need to apply the rule of negative exponents and simplify the resulting expression. This involves rewriting the expression with a positive exponent and then evaluating the exponent.

Q: What is the final value of the expression $ \frac{1}{4^{-2}} $?

A: The final value of the expression $ \frac{1}{4^{-2}} $ is 16. This is because $ 4^{-2} = \frac{1}{4^2} = \frac{1}{16} $, and the reciprocal of $ \frac{1}{16} $ is 16.

Q: Can I simplify an expression with a negative exponent by multiplying the numerator and denominator by the same value?

A: Yes, you can simplify an expression with a negative exponent by multiplying the numerator and denominator by the same value. For example, to simplify the expression $ \frac{1}{2^{-3}} $, you can multiply the numerator and denominator by $ 2^3 $ to get:

$ \frac{1}{2^{-3}} = \frac{2^3}{2{-3} \times 2^3} = \frac{2^3}{1} = 8 $

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

A: To evaluate an expression with a negative exponent and a variable, you need to apply the rule of negative exponents and simplify the resulting expression. This involves rewriting the expression with a positive exponent and then evaluating the exponent.

Q: Can I use the rule of negative exponents to simplify an expression with a negative exponent and a fraction?

A: Yes, you can use the rule of negative exponents to simplify an expression with a negative exponent and a fraction. For example, to simplify the expression $ \frac{1}{\frac{1}{2^{-3}}} $, you can apply the rule of negative exponents and simplify the resulting expression:

$ \frac{1}{\frac{1}{2^{-3}}} = \frac{1}{\frac{1}{8}} = 8 $

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

A: To evaluate an expression with a negative exponent and a decimal, you need to apply the rule of negative exponents and simplify the resulting expression. This involves rewriting the expression with a positive exponent and then evaluating the exponent.