Simplifying Exponential Expressions A Step By Step Guide

In the realm of mathematics, simplifying expressions is a fundamental skill. It allows us to take complex equations and reduce them to their most basic form, making them easier to understand and manipulate. This article delves into the process of simplifying expressions with exponents, focusing on the specific example of $(-7)^{10}(-7)^{14}$. We will explore the underlying principles, step-by-step methods, and provide a comprehensive explanation to ensure clarity and mastery of this concept.

Understanding Exponents

Before we dive into the specific problem, let's first establish a solid understanding of exponents. An exponent represents the number of times a base is multiplied by itself. For example, in the expression $a^n$, 'a' is the base and 'n' is the exponent. This means 'a' is multiplied by itself 'n' times. So, $2^3$ means 2 multiplied by itself 3 times (2 * 2 * 2 = 8).

The Product of Powers Rule

When multiplying exponents with the same base, there's a fundamental rule we can apply, known as the Product of Powers Rule. This rule states that when multiplying powers with the same base, we add the exponents. Mathematically, it's expressed as: $a^m * a^n = a^{m+n}$. This rule is the cornerstone of simplifying the given expression. It streamlines the process and allows us to combine terms efficiently. Applying this rule correctly is crucial for accurate simplification.

Applying the Product of Powers Rule to $(-7)^{10}(-7)^{14}$

Now, let's apply this rule to our problem: $(-7)^{10}(-7)^{14}$. We observe that both terms have the same base, which is -7. According to the Product of Powers Rule, we can add the exponents:

$(-7)^{10}(-7)^{14} = (-7)^{10+14}$

Simplifying the Exponent

Next, we simply add the exponents: 10 + 14 = 24. So, our expression becomes:

$(-7)^{24}$

This is the simplified form of the expression. We have successfully combined the two terms into a single term with a single exponent.

Why We Don't Expand Further

The instruction explicitly states: "Do not expand or simplify further." This instruction is crucial. While we could calculate the actual numerical value of $(-7)^{24}$, doing so would violate the given instructions. The purpose of this exercise is to demonstrate the application of the Product of Powers Rule, not to compute large numerical values. Expanding $(-7)^{24}$ would result in a very large number, and the simplified form $(-7)^{24}$ is more concise and mathematically elegant.

Understanding the Significance of a Negative Base

It's important to note the significance of the negative base (-7) in this expression. When a negative number is raised to an even power, the result is always positive. This is because the negative signs cancel out in pairs during multiplication. For example, $(-2)^2 = (-2) * (-2) = 4$. In our case, $(-7)^{24}$ will be a positive number because 24 is an even exponent. However, we are not asked to calculate this value, but rather to simplify the expression using exponent rules.

Step-by-Step Breakdown

To recap, here's a step-by-step breakdown of how we simplified the expression:

1. Identify the Common Base: We observed that both terms in the expression $(-7)^{10}(-7)^{14}$ have the same base, which is -7.
2. Apply the Product of Powers Rule: We used the rule $a^m * a^n = a^{m+n}$ to rewrite the expression as $(-7)^{10+14}$.
3. Add the Exponents: We added the exponents 10 and 14 to get 24, resulting in $(-7)^{24}$.
4. Final Simplified Form: We presented $(-7)^{24}$ as the simplified form, adhering to the instruction not to expand or simplify further.

Common Mistakes to Avoid

When simplifying expressions with exponents, it's essential to avoid common mistakes. Here are a few to keep in mind:

• Incorrectly Applying the Product of Powers Rule: Ensure you only add exponents when the bases are the same. For example, $2^3 * 3^2$ cannot be simplified using the Product of Powers Rule because the bases (2 and 3) are different.
• Forgetting the Sign with Negative Bases: Pay close attention to the sign when dealing with negative bases. Remember that a negative base raised to an even power results in a positive number, while a negative base raised to an odd power results in a negative number.
• Expanding When Not Required: Always follow the instructions carefully. In this case, we were explicitly told not to expand further, even though we could calculate the numerical value of $(-7)^{24}$.
• Confusing Addition and Multiplication: The Product of Powers Rule applies to multiplication of exponents with the same base, not addition. $a^m + a^n$ cannot be simplified using this rule.

Importance of Simplifying Expressions

Simplifying expressions is a crucial skill in mathematics for several reasons:

• Clarity: Simplified expressions are easier to understand and interpret. They present the information in a concise and organized manner.
• Problem-Solving: Simplified expressions make it easier to solve equations and manipulate formulas. They reduce the complexity of calculations.
• Efficiency: Simplifying expressions can save time and effort. Working with a simplified expression is often much faster than working with a complex one.
• Foundation for Advanced Concepts: Simplifying expressions is a foundational skill for more advanced mathematical concepts, such as algebra, calculus, and beyond.

Conclusion

In conclusion, simplifying expressions with exponents involves applying specific rules and techniques to reduce the expression to its most basic form. In the case of $(-7)^{10}(-7)^{14})$, we successfully applied the Product of Powers Rule to obtain the simplified form $(-7)^{24}$. It's crucial to understand the underlying principles, follow instructions carefully, and avoid common mistakes. Mastering these skills will significantly enhance your mathematical abilities and pave the way for success in more advanced topics. This exercise highlights the importance of adhering to instructions and focusing on the specific skill being assessed, which in this case was the application of the Product of Powers Rule. By understanding and applying these rules, you can confidently simplify a wide range of expressions with exponents.