Solving Exponential Variables: Find A, B, And C

Hey guys! Let's dive into this math problem where we need to figure out the values of the variables a, b, and c in a simplified exponential expression. It looks a little intimidating at first, but we'll break it down step by step so it's super easy to understand. We’re dealing with the expression $\frac{2^5 \cdot 8^4}{16}=\frac{2^5 \cdot(2^a)^4}{2^4}=\frac{2^5 \cdot 2^b}{2^4}=2^c$, and our mission is to find out what a, b, and c are. Ready? Let's get started!

Breaking Down the Problem

Before we jump into solving for the variables, let's take a good look at the expression we've got:

$\frac{2^5 \cdot 8^4}{16}=\frac{2^5 \cdot(2^a)^4}{2^4}=\frac{2^5 \cdot 2^b}{2^4}=2^c$

Our main goal here is to simplify this expression by using the properties of exponents. Remember, the key to making these problems easier is to express everything in terms of the same base. In this case, our base is 2, which is super convenient! We're going to use a few exponent rules, so let's quickly refresh those:

1. Power of a Power: $(x^m)^n = x^{m \cdot n}$
2. Product of Powers: $x^m \cdot x^n = x^{m+n}$
3. Quotient of Powers: $\frac{x^m}{x^n} = x^{m-n}$

With these rules in our toolkit, we'll be able to simplify the expression and find the values of a, b, and c. Let's start by tackling the first part of the equation and converting everything to a base of 2.

Step-by-Step Solution

Finding the Value of 'a'

Okay, let's start by focusing on the first part of the equation:

$\frac{2^5 \cdot 8^4}{16}=\frac{2^5 \cdot(2^a)^4}{2^4}$

The first thing we need to do is express 8 and 16 as powers of 2. We know that $8 = 2^3$ and $16 = 2^4$. So, we can rewrite the equation as:

$\frac{2^5 \cdot (2^3)^4}{2^4} = \frac{2^5 \cdot (2^a)^4}{2^4}$

Now, let’s use the power of a power rule, which states that $(x^m)^n = x^{m \cdot n}$. Applying this to $(2^3)^4$, we get $2^{3 \cdot 4} = 2^{12}$. So, the left side of the equation becomes:

$\frac{2^5 \cdot 2^{12}}{2^4} = \frac{2^5 \cdot (2^a)^4}{2^4}$

Now we can see that $(2^a)^4$ should be equal to $2^{12}$. Again using the power of a power rule, we know that $(2^a)^4 = 2^{4a}$. So, we have:

$2^{4a} = 2^{12}$

Since the bases are the same, the exponents must be equal. This gives us the equation:

$4a = 12$

To solve for 'a', we simply divide both sides by 4:

$a = \frac{12}{4}$

$a = 3$

So, we've found our first variable! a = 3. Awesome, right? Now, let's move on to finding the value of 'b'.

Determining the Value of 'b'

Moving on, let's find the value of 'b'. We have the expression:

$\frac{2^5 \cdot (2^a)^4}{2^4} = \frac{2^5 \cdot 2^b}{2^4}$

We already know that $a = 3$, so we can substitute that into the equation:

$\frac{2^5 \cdot (2^3)^4}{2^4} = \frac{2^5 \cdot 2^b}{2^4}$

We've already simplified $(2^3)^4$ to $2^{12}$, so the equation becomes:

$\frac{2^5 \cdot 2^{12}}{2^4} = \frac{2^5 \cdot 2^b}{2^4}$

Now, let’s focus on the numerators. We can use the product of powers rule, which states that $x^m \cdot x^n = x^{m+n}$. So, $2^5 \cdot 2^{12} = 2^{5+12} = 2^{17}$. Our equation now looks like this:

$\frac{2^{17}}{2^4} = \frac{2^5 \cdot 2^b}{2^4}$

To make things even clearer, let's multiply both sides of the equation by $2^4$ to get rid of the denominators:

$2^{17} = 2^5 \cdot 2^b$

Now, we need to figure out what exponent we need to add to 5 to get 17. In other words, we need to use the product of powers rule in reverse. We can rewrite the equation as:

$2^{17} = 2^{5+b}$

Since the bases are the same, the exponents must be equal:

$17 = 5 + b$

To solve for 'b', subtract 5 from both sides:

$b = 17 - 5$

$b = 12$

Great! We've found that b = 12. Only one variable left to go – let's find 'c'!

Calculating the Value of 'c'

Alright, let's wrap this up by finding the value of 'c'. We have the equation:

$\frac{2^5 \cdot 2^b}{2^4} = 2^c$

We already know that $b = 12$, so we can substitute that into the equation:

$\frac{2^5 \cdot 2^{12}}{2^4} = 2^c$

Again, let's simplify the numerator using the product of powers rule: $2^5 \cdot 2^{12} = 2^{5+12} = 2^{17}$. So, the equation becomes:

$\frac{2^{17}}{2^4} = 2^c$

Now we use the quotient of powers rule, which says that $\frac{x^m}{x^n} = x^{m-n}$. Applying this, we get:

$2^{17-4} = 2^c$

$2^{13} = 2^c$

Since the bases are the same, the exponents must be equal:

$13 = c$

So, we've found that c = 13. We did it!

Final Values

We've successfully found the values for a, b, and c:

• a = 3
• b = 12
• c = 13

To recap, we started with the expression $\frac{2^5 \cdot 8^4}{16}=\frac{2^5 \cdot(2^a)^4}{2^4}=\frac{2^5 \cdot 2^b}{2^4}=2^c$ and used the properties of exponents to simplify and solve for each variable. By converting all terms to a base of 2 and applying the power of a power, product of powers, and quotient of powers rules, we were able to break down the problem and find the solutions.

I hope this step-by-step explanation made it super clear how to solve for these exponential variables. Remember, guys, the key is to take it one step at a time and use those exponent rules! You've got this!