Simplifying Expressions Finding A And B In X^a Y^b

Hey there, math enthusiasts! Today, we're diving into a fun algebraic simplification problem. Our mission? To break down the expression $\frac{x^2(y3)^4}{xy5}$ and rewrite it in the form $x^a \cdot y^b$. This means we need to figure out the values of 'a' and 'b'. Don't worry, we'll take it step by step, making sure everyone can follow along. Let's get started and unlock the secrets of exponents and algebraic manipulation!

Initial Expression and Our Goal

Before we roll up our sleeves and begin the simplification journey, let's take a moment to clearly state the expression we are working with, and what our ultimate goal is. The initial expression is: $\frac{x^2(y3)^4}{xy5}$ Our main objective here is to manipulate this algebraic expression by applying the rules of exponents, simplifying it until it perfectly fits into the form: $x^a \cdot y^b$ In this final form, 'a' will represent the exponent of 'x' and 'b' will represent the exponent of 'y'. It's like we're on a quest to find these two values! So, with our expression clearly defined and our goal set, let's dive into the first steps of simplification. Are you ready to see some exponent magic?

Step 1: Power to a Power

The first rule of exponents we're going to use is the "power to a power" rule. This rule states that when you raise a power to another power, you multiply the exponents. Mathematically, it looks like this: $(x^m)n = x^{m \cdot n}$ In our expression, we have the term $(y³⁾4$. Applying the power to a power rule, we multiply the exponents 3 and 4. This gives us:

$$(y^3)^4 = y^{3 \cdot 4} = y^{12}$$

Now, let's substitute this back into our original expression. Our expression now looks like this:

$$\frac{x^2 y^{12}}{xy^5}$$

See how we've already made progress? By applying just one rule, we've simplified a part of our expression. Next, we'll tackle the division part, which will bring us even closer to our final form. Keep your eyes on the exponents; they're the key to solving this puzzle!

Step 2: Dividing with the Same Base

Now, let's handle the division in our expression. When we divide terms with the same base, we subtract the exponents. This rule can be expressed as: $\frac{x^m}{xn} = x^{m-n}$ In our simplified expression, $\frac{x^2 y^{12}}{xy5}$, we have two divisions to consider: one for the 'x' terms and one for the 'y' terms.

For the 'x' terms, we have $\fracx^2}{x}$. Remember, if a variable doesn't have an exponent written, it's understood to be 1. So, we have $\frac{x^2}{x1}$. Applying the rule, we subtract the exponents $x^{2-1 = x^1 = x$

For the 'y' terms, we have $\fracy^{12}}{y5}$. Again, we subtract the exponents $y^{12-5 = y^7$

Now, let's put these simplified terms back together. Our expression now looks like this:

$$x y^7$$

Wow, we're getting so close! We've successfully simplified the expression using the division rule of exponents. Next up, we'll see how this simplified form relates to our target form of $x^a \cdot y^b$, and we'll finally unveil the values of 'a' and 'b'. Exciting, isn't it?

Step 3: Identifying a and b

Alright, guys, we've reached the final stage of our simplification journey! Our expression is now beautifully simplified to: $x y^7$ Our initial goal was to rewrite the given expression in the form $x^a \cdot y^b$. Now, let's compare what we have with what we want. It's like matching puzzle pieces!

We have $x y^7$, which can also be written as $x^1 y^7$. This is because, as we discussed earlier, if a variable doesn't have an explicit exponent, it's understood to be 1.

Now, we line it up with our target form: $x^a \cdot y^b$

Can you see it? It's like looking for a familiar face in a crowd. By comparing the exponents, we can directly identify the values of 'a' and 'b'.

For the 'x' term, we have $x^1$, which corresponds to $x^a$. Therefore, $a = 1$

For the 'y' term, we have $y^7$, which corresponds to $y^b$. Therefore, $b = 7$

And there you have it! We've successfully navigated the world of exponents and simplified our expression. But, before we celebrate our victory, let's clearly state our final answer.

Final Answer: a = 1, b = 7

After all our hard work and simplifying steps, we've arrived at the solution! We found that when we simplify the expression $\frac{x^2(y3)^4}{xy5}$ and rewrite it in the form $x^a \cdot y^b$, the values of 'a' and 'b' are:

$$a = 1$$

$$b = 7$$

So, the final simplified form of our expression is $x^1 y^7$, or simply $x y^7$.

Great job, everyone! We tackled this algebraic problem head-on, applying the rules of exponents like pros. Remember, the key to simplifying expressions is to take it one step at a time, applying the relevant rules as you go. And now, we can confidently say we've conquered this challenge. What a fantastic journey through the world of algebra!