Evaluating The Expression (-4)^-3 A Step-by-Step Guide

In this article, we will delve into the process of evaluating the expression $(-4)^{-3}$. This involves understanding the rules of exponents, particularly how to handle negative exponents. By breaking down the problem step-by-step, we will arrive at the solution and provide a clear explanation of the underlying concepts.

Understanding Negative Exponents

The key concept here is the negative exponent. A negative exponent indicates that we need to take the reciprocal of the base raised to the positive version of the exponent. In other words, $a^{-n} = \frac{1}{a^n}$. This rule is crucial for simplifying expressions with negative exponents, and it's the foundation for solving our problem.

When dealing with negative exponents, it's important to remember that the negative sign does not make the number negative. Instead, it signifies a reciprocal. This is a common point of confusion, so let's emphasize it: a negative exponent means we are dealing with a fraction, where the base raised to the positive exponent is in the denominator.

Moreover, understanding how negative exponents interact with negative bases is essential. In our case, the base is -4, and the exponent is -3. We need to apply the negative exponent rule carefully, considering the implications of raising a negative number to a power. A negative number raised to an odd power will result in a negative number, while a negative number raised to an even power will result in a positive number. This detail will be critical as we proceed with the calculation.

Step 1: Apply the Negative Exponents Rule

Our initial expression is $(-4)^{-3}$. To apply the negative exponents rule, we rewrite this as a fraction with 1 as the numerator and $(-4)^3$ as the denominator. This gives us:

$$\frac{1}{(-4)^3}$$

This transformation is the heart of simplifying expressions with negative exponents. By taking the reciprocal, we convert the negative exponent into a positive one, making the expression easier to evaluate. This step is not just a mechanical application of a rule; it's a fundamental shift in how we perceive the expression. Instead of dealing with a negative power, we are now dealing with a fraction, which is often more intuitive to handle.

The significance of this step cannot be overstated. It allows us to move from a potentially confusing expression to a more manageable form. The negative exponent, which might seem intimidating at first, is now transformed into a simple reciprocal. This is a powerful technique in algebra, and it's one that you'll use repeatedly in various mathematical contexts. By understanding this step thoroughly, you'll gain a significant advantage in simplifying complex expressions.

Step 2: Expand the Exponent

Now that we have $\frac{1}{(-4)^3}$, we need to expand the exponent. This means writing out the base, -4, multiplied by itself three times, as indicated by the exponent 3. This gives us:

$$\frac{1}{(-4)(-4)(-4)}$$

Expanding the exponent helps us visualize the multiplication process. It breaks down the power into a series of simple multiplications, making it easier to calculate the final result. This step is particularly helpful when dealing with exponents that are not immediately obvious, such as larger numbers or variables.

By writing out the repeated multiplication, we are essentially unpacking the exponent. We are making explicit the operation that the exponent represents. This can be especially useful when dealing with negative bases, as it helps to keep track of the signs. In our case, we have three negative numbers being multiplied together, which will result in a negative product.

This step also highlights the importance of understanding the definition of an exponent. An exponent tells us how many times to multiply the base by itself. Expanding the exponent makes this definition concrete and tangible, preventing potential errors in calculation. It's a step that bridges the gap between abstract notation and concrete arithmetic.

Step 3: Simplify

Finally, we simplify the expression. We multiply the numbers in the denominator: $(-4)(-4)(-4)$.

First, multiply the first two -4s: $(-4)(-4) = 16$. A negative times a negative is a positive.

Then, multiply the result by the remaining -4: $(16)(-4) = -64$. A positive times a negative is a negative.

So, the denominator simplifies to -64. Therefore, the entire expression becomes:

$$\frac{1}{-64}$$

This is our final simplified answer. We have successfully evaluated $(-4)^{-3}$ by applying the rules of exponents and performing the necessary arithmetic operations.

The simplification step is where all the previous steps come together. It's the culmination of our efforts to rewrite and expand the expression. By carefully multiplying the numbers in the denominator, we arrive at a single value that represents the entire expression. This step requires attention to detail, especially when dealing with negative numbers. A single sign error can change the entire result.

Moreover, the simplification step reinforces the concept of order of operations. We first dealt with the exponent, then performed the multiplication. This order is crucial for arriving at the correct answer. By consistently following the order of operations, we can avoid common errors and ensure that our calculations are accurate.

The Final Result

The value of $(-4)^{-3}$ is $\frac{1}{-64}$, which can also be written as $-\frac{1}{64}$.

This final answer is the result of a systematic application of the rules of exponents and careful arithmetic. We started with a seemingly complex expression and, through a series of steps, transformed it into a simple fraction. This process highlights the power of mathematical rules and the importance of step-by-step problem-solving.

Understanding the negative exponent rule is crucial for working with various mathematical concepts, including scientific notation, algebraic manipulations, and calculus. This example provides a solid foundation for tackling more complex problems involving exponents. By mastering these fundamental principles, you'll be well-equipped to handle a wide range of mathematical challenges.

Furthermore, this example demonstrates the importance of precision in mathematical calculations. A small error in any step can lead to a completely different result. By paying attention to detail and carefully checking our work, we can ensure that our answers are accurate and reliable. This meticulous approach is essential for success in mathematics and other quantitative fields.

Conclusion

In conclusion, we have successfully evaluated the expression $(-4)^{-3}$ by applying the negative exponents rule, expanding the exponent, and simplifying the result. The final answer is $-\frac{1}{64}$. This exercise demonstrates the importance of understanding and applying the rules of exponents, as well as the value of breaking down complex problems into smaller, more manageable steps. By mastering these techniques, you can confidently tackle a wide range of mathematical challenges.

This step-by-step approach not only leads to the correct answer but also provides a deeper understanding of the underlying mathematical principles. Each step builds upon the previous one, creating a logical progression that reinforces the concepts involved. This is a valuable strategy for learning mathematics, as it encourages critical thinking and problem-solving skills.

Moreover, this example highlights the interconnectedness of mathematical concepts. The negative exponent rule is not an isolated concept; it's connected to other rules of exponents, as well as to the fundamental operations of arithmetic. By understanding these connections, we can gain a more holistic view of mathematics and its applications. This holistic understanding is crucial for developing mathematical fluency and confidence.

Therefore, the value of the expression $(-4)^{-3}$ is $-\frac{1}{64}$.