Simplify Exponential Expressions And Evaluate At A = -5 A Step-by-Step Guide

Hey guys! Today, we're diving deep into the world of exponents. Exponents might seem intimidating at first, but trust me, once you grasp the basic properties, they become super easy to handle. We're going to break down an expression step by step, simplify it using exponent rules, and then find its value for a given variable. So, buckle up and let's get started!

Understanding the Expression

Let's tackle the expression at hand: $\frac{\left(2 a^4\right)^2}{a^2 a^3}$. This looks a bit complex, right? But don't worry, we'll simplify it piece by piece. Our main goal here is to use the properties of exponents to rewrite this expression in its simplest form. This involves applying several key rules that govern how exponents behave when dealing with multiplication, division, and powers. By understanding these rules, you can transform complicated expressions into much simpler forms, making them easier to evaluate and work with. The process of simplification not only makes the expression more manageable but also reveals the underlying structure and relationships between the variables and constants involved. So, let's dive in and start breaking down this expression to see how we can make it simpler and more understandable.

Breaking Down the Numerator

First, let's focus on the numerator: $(2a^4)^2$. Here, we have a product raised to a power. Remember the power of a product rule? It states that $(ab)^n = a^n b^n$. Applying this rule, we get:

$(2a^4)^2 = 2^2 * (a^4)^2$

Now, we have another exponent rule to use: the power of a power rule, which says $(a^m)^n = a^{m*n}$. Applying this to our expression, we get:

$2^2 * (a^4)^2 = 4 * a^{4*2} = 4a^8$

So, we've successfully simplified the numerator to $4a^8$. This step is crucial because it transforms a seemingly complex term into a more manageable one. The power of a product rule allows us to distribute the exponent outside the parenthesis to each factor inside, while the power of a power rule simplifies nested exponents by multiplying them. By mastering these rules, you'll find that simplifying expressions becomes second nature. This is not just about getting the right answer; it's about understanding the mechanics of exponents and how they interact with each other. Think of each step as peeling back a layer, revealing the simpler form underneath. With practice, you'll become adept at identifying which rule to apply in any given situation, making even the most daunting expressions seem less intimidating.

Simplifying the Denominator

Next up, let's simplify the denominator: $a^2 a^3$. This is a straightforward application of the product of powers rule, which states that $a^m * a^n = a^{m+n}$. So, we have:

$a^2 a^3 = a^{2+3} = a^5$

Alright, the denominator is now simplified to $a^5$. The product of powers rule is one of the most fundamental rules when dealing with exponents. It tells us that when we multiply terms with the same base, we can simply add their exponents. This rule is not just a shortcut; it reflects the basic definition of exponents as repeated multiplication. For example, $a^2$ means $a * a$, and $a^3$ means $a * a * a$. When we multiply $a^2$ and $a^3$, we are essentially multiplying $(a * a) * (a * a * a)$, which gives us $a^5$ or $a * a * a * a * a$. Understanding the 'why' behind the rule makes it easier to remember and apply. Simplifying the denominator is a crucial step because it sets us up for the next phase, which involves dividing the simplified numerator by the simplified denominator. The simpler each part is, the easier the overall simplification process becomes.

Combining and Simplifying the Fraction

Now that we've simplified both the numerator and the denominator, let's put them back together. Our expression now looks like this:

$\frac{4a^8}{a^5}$

Here, we can use the quotient of powers rule, which states that $\frac{a^m}{a^n} = a^{m-n}$. Applying this rule, we get:

$\frac{4a^8}{a^5} = 4a^{8-5} = 4a^3$

Yay! We've successfully rewritten the expression to its simplest form: $4a^3$. The quotient of powers rule is the counterpart to the product of powers rule and is equally essential for simplifying expressions. It's based on the idea that when you divide terms with the same base, you're essentially canceling out factors. For instance, if we were to expand $a^8$ and $a^5$, we'd have eight 'a's in the numerator and five 'a's in the denominator. When we divide, five of those 'a's cancel out, leaving us with three 'a's in the numerator, which is represented as $a^3$. The constant '4' remains untouched because it doesn't have a corresponding term to cancel out with in the denominator. This simplified form, $4a^3$, is much easier to work with than our original expression, especially when we need to evaluate it for a specific value of 'a'.

Evaluating the Expression

Now, the final part of the question asks us to find the value of the rewritten expression when $a = -5$. So, we substitute $a$ with $-5$ in our simplified expression:

$4a^3 = 4(-5)^3$

First, we calculate $(-5)^3$:

$(-5)^3 = -5 * -5 * -5 = -125$

Then, we multiply by 4:

$4 * -125 = -500$

Therefore, the value of the rewritten expression when $a = -5$ is $-500$.

Substitution is a key skill in algebra and beyond. Once we've simplified an expression, substituting a given value for a variable allows us to find the numerical result. In this case, we replaced 'a' with '-5' in our simplified expression $4a^3$. Remember the order of operations (PEMDAS/BODMAS): we first handle the exponent, then the multiplication. So, we calculated $(-5)^3$, which means -5 multiplied by itself three times. This gave us -125. Then, we multiplied -125 by 4 to get our final answer of -500. It's crucial to pay attention to the sign when dealing with negative numbers raised to powers. A negative number raised to an odd power will always result in a negative number, while a negative number raised to an even power will result in a positive number. This detail is essential for accurate calculations and problem-solving in mathematics.

Conclusion

So, the correct answer is D. -500. We took a complex expression, simplified it using the properties of exponents, and then evaluated it for a given value. Remember, guys, the key to mastering exponents is practice and understanding the rules. Keep practicing, and you'll become exponent pros in no time! By walking through this problem step-by-step, we've not only found the solution but also reinforced the importance of understanding and applying exponent rules. The journey from the initial complex expression to the final numerical answer showcases how mathematical concepts build upon each other. Each rule we've used, from the power of a product to the quotient of powers, plays a crucial role in simplifying and solving the problem. This process of simplification is not just about finding the answer; it's about developing a deeper understanding of algebraic structures and how they behave. So, next time you encounter an expression with exponents, remember the principles we've discussed here, and tackle it with confidence!

Rewrite the expression $\frac{\left(2 a^4\right)^2}{a^2 a^3}$ using the properties of exponents. What is the value of the rewritten expression when $a=-5$?

Simplify Exponential Expressions and Evaluate at a = -5 A Step-by-Step Guide