Solving $(-5)^2 \div (-5)^{-3}$ A Step-by-Step Guide

Hey there, math enthusiasts! Today, we're going to embark on an exciting journey to solve a seemingly complex mathematical expression: $(-5)^2 \div (-5)^{-3}$. This problem beautifully combines the concepts of exponents and division, and by the end of this exploration, you'll not only be able to solve this particular problem but also gain a deeper understanding of these fundamental mathematical principles. So, buckle up and let's dive in!

Understanding the Building Blocks: Exponents

Before we tackle the main problem, let's first solidify our understanding of exponents. In simple terms, an exponent indicates how many times a number, called the base, is multiplied by itself. For instance, in the expression $a^n$, 'a' is the base, and 'n' is the exponent. This means we multiply 'a' by itself 'n' times. Think of it as a shorthand for repeated multiplication.

For example, $2^3$ means 2 multiplied by itself 3 times, which is $2 \* 2 \* 2 = 8$. Similarly, $5^2$ means 5 multiplied by itself 2 times, resulting in $5 \* 5 = 25$. Now, let's consider negative exponents. A negative exponent indicates the reciprocal of the base raised to the positive value of the exponent. Mathematically, $a^{-n} = \frac{1}{a^n}$. This means that $(-5)^{-3}$ is the same as $\frac{1}{(-5)^3}$. This understanding is crucial for solving our main problem.

Delving Deeper into Negative Bases and Exponents

Now, let's add another layer of complexity by considering negative bases. When a negative number is raised to an exponent, the outcome depends on whether the exponent is even or odd. If the exponent is even, the result is positive because a negative number multiplied by itself an even number of times yields a positive result. For example, $(-2)^2 = (-2) \* (-2) = 4$. On the other hand, if the exponent is odd, the result is negative because a negative number multiplied by itself an odd number of times results in a negative number. For instance, $(-2)^3 = (-2) \* (-2) \* (-2) = -8$. These rules are essential to keep in mind as we move forward.

Dividing Powers with the Same Base

Now that we have a solid grasp of exponents, let's shift our focus to division. When dividing powers with the same base, there's a nifty rule that simplifies the process. The rule states that $a^m \div a^n = a^{m-n}$. In other words, when dividing powers with the same base, we subtract the exponents. This rule is a direct consequence of the definition of exponents and the properties of fractions. To illustrate, let's consider a simple example: $3^5 \div 3^2$. According to the rule, this is equal to $3^{5-2} = 3^3 = 27$. This rule is a powerful tool that will help us simplify our original problem.

A Practical Example

Let's consider another example to solidify this concept. Imagine we have $4^4 \div 4^2$. Applying the rule, we subtract the exponents: $4^{4-2} = 4^2 = 16$. We can also think of this in terms of repeated multiplication: $\frac{4 \* 4 \* 4 \* 4}{4 \* 4}$. We can cancel out two 4s from the numerator and denominator, leaving us with $4 \* 4 = 16$. This demonstrates the underlying principle behind the rule.

Tackling the Problem: $(-5)^2 \div (-5)^{-3}$

Alright, guys, now we're fully equipped to tackle our main problem: $(-5)^2 \div (-5)^{-3}$. Let's break it down step by step. First, we can apply the rule for dividing powers with the same base, which states that $a^m \div a^n = a^{m-n}$. In our case, the base is -5, m is 2, and n is -3. So, we have $(-5)^2 \div (-5)^{-3} = (-5)^{2 - (-3)}$.

Simplifying the Exponent

The next step is to simplify the exponent. We have $2 - (-3)$, which is the same as $2 + 3 = 5$. Therefore, our expression becomes $(-5)^5$. Now, we need to evaluate $(-5)^5$. As we discussed earlier, when a negative number is raised to an odd exponent, the result is negative. So, we know that $(-5)^5$ will be a negative number.

Calculating the Final Result

To find the value, we multiply -5 by itself five times: $(-5)^5 = (-5) \* (-5) \* (-5) \* (-5) \* (-5)$. Let's do this step by step:

• $(-5) \* (-5) = 25$
• $25 \* (-5) = -125$
• $-125 \* (-5) = 625$
• $625 \* (-5) = -3125$

Therefore, $(-5)^5 = -3125$. So, the solution to our problem, $(-5)^2 \div (-5)^{-3}$, is -3125. Congratulations, we've successfully navigated this mathematical challenge!

Alternative Approach: Dealing with Negative Exponents

There's also another way to approach this problem, which involves dealing with the negative exponent directly. Remember that $a^{-n} = \frac{1}{a^n}$. So, we can rewrite $(-5)^{-3}$ as $\frac{1}{(-5)^3}$. This means our original expression, $(-5)^2 \div (-5)^{-3}$, can be rewritten as $(-5)^2 \div \frac{1}{(-5)^3}$.

Rewriting the Division as Multiplication

Dividing by a fraction is the same as multiplying by its reciprocal. Therefore, we can rewrite $(-5)^2 \div \frac{1}{(-5)^3}$ as $(-5)^2 \* (-5)^3$. Now, we have a product of powers with the same base. The rule for multiplying powers with the same base states that $a^m \* a^n = a^{m+n}$. Applying this rule, we get $(-5)^{2+3} = (-5)^5$.

Arriving at the Same Destination

As we saw in our previous approach, $(-5)^5 = -3125$. So, regardless of the method we choose, we arrive at the same answer. This highlights the beauty of mathematics – there are often multiple paths to the same solution!

Key Takeaways and Conclusion

Wow, what a journey! We've successfully solved the problem $(-5)^2 \div (-5)^{-3}$ and, more importantly, gained a deeper understanding of exponents and division. We've learned about the rules for dividing powers with the same base, the significance of negative exponents, and how to handle negative bases raised to different powers. Remember, practice is key to mastering these concepts.

Final Thoughts

Math can sometimes seem intimidating, but by breaking down complex problems into smaller, manageable steps, we can conquer any challenge. Keep exploring, keep questioning, and keep practicing. The world of mathematics is full of fascinating discoveries waiting to be made. I hope this explanation has been helpful and has sparked your curiosity to delve even deeper into the world of numbers and equations. Keep up the great work, guys, and I'll see you in the next mathematical adventure!