Simplify (1/5)p^(2/3)^2 * (1/4)p^(3/2)^2 / (1/5)p^(1/3)^2

Hey guys! Let's dive into simplifying a tricky algebraic expression. We've got a doozy here: seperlima p ^ 2/3^2 * 1/4 p ^ 3/2 ^ 2 / 1/5 P pangkat sepertiga pangkat 2. Sounds intimidating, right? But don't worry, we'll break it down piece by piece, making it super easy to understand. Our goal is to make algebraic expressions less scary and more like a fun puzzle. We'll go through each step slowly and carefully, so you can follow along and learn how to tackle these problems yourself. Think of it as unlocking a secret code – once you know the rules, you can decipher any expression! So, grab your pencils and let's get started on this mathematical adventure!

Understanding the Expression

Alright, let’s first understand what we're dealing with. The expression is: seperlima p ^ 2/3^2 * 1/4 p ^ 3/2 ^ 2 / 1/5 P pangkat sepertiga pangkat 2. We need to translate this into standard mathematical notation to make sense of it. Keywords such as 'seperlima' which means one-fifth (1/5), 'pangkat' meaning power, and fractions will guide us. Rewriting it, we get: (1/5) * (p^(2/3))2 * (1/4) * (p^(3/2))2 / (1/5) * (p^(1/3))2. Now it looks more manageable, right? The key here is recognizing the different parts and how they connect. We have fractions, variables with exponents, and multiplication and division operations. Understanding each component is crucial before we start simplifying. Think of it like reading a map – you need to know the symbols and directions before you can find your way. In this case, the mathematical symbols and operations are our map, guiding us to the simplified answer. So, let's keep this translated form in mind as we move forward. We'll be using it as our foundation for the next steps.

Breaking Down the Components

Let's break down each component of the expression. First, we have the fractions: 1/5 and 1/4. These are constants that we'll deal with later. Then we have the variable 'p' raised to different powers. These powers are fractions themselves: 2/3, 3/2, and 1/3. Understanding exponents is key here. Remember, an exponent tells you how many times to multiply the base by itself. But when the exponent is a fraction, it represents a root. For example, p^(1/2) is the square root of p, and p^(1/3) is the cube root of p. Now, we also have the expression (p^(2/3))2. This means we're raising p^(2/3) to the power of 2. Remember the rule of exponents: (a^m)n = a^(m*n). We'll use this rule to simplify this part. Similarly, we have (p^(3/2))2 and (p^(1/3))2. We'll apply the same rule to simplify them. By understanding each component individually, we can see how they fit together in the larger expression. It's like understanding the individual instruments in an orchestra before listening to the whole symphony. Each part plays a crucial role, and knowing what each does helps us appreciate the overall complexity and beauty of the expression. So, let’s hold onto this component-by-component understanding as we move on to the next step: applying the exponent rules.

Applying Exponent Rules

Now, let’s apply some exponent rules to simplify the powers of 'p'. We have (p^(2/3))2, (p^(3/2))2, and (p^(1/3))2. Remember the rule: (a^m)n = a^(m*n). So, (p^(2/3))2 becomes p^((2/3)*2) = p^(4/3). Similarly, (p^(3/2))2 becomes p^((3/2)*2) = p^3. And (p^(1/3))2 becomes p^((1/3)*2) = p^(2/3). See how much simpler it looks now? By applying this one rule, we've eliminated the parentheses and combined the exponents. Now our expression looks like this: (1/5) * p^(4/3) * (1/4) * p^3 / (1/5) * p^(2/3). Next, we need to deal with the multiplication and division of these terms. Remember, when multiplying terms with the same base, we add the exponents: a^m * a^n = a^(m+n). And when dividing, we subtract the exponents: a^m / a^n = a^(m-n). These rules are like the secret sauce for simplifying expressions with exponents. Once you master them, you can tackle even the trickiest problems. So, let’s keep these rules in mind as we move on to the next step: combining like terms. We're making great progress, guys! The expression is slowly but surely transforming into a simpler form.

Combining Like Terms

Time to combine those like terms! Our expression is now: (1/5) * p^(4/3) * (1/4) * p^3 / (1/5) * p^(2/3). First, let’s deal with the constants. We have (1/5) * (1/4) in the numerator and (1/5) in the denominator. So, the constants part of the expression is (1/5) * (1/4) / (1/5). This simplifies to (1/20) / (1/5), which is the same as (1/20) * (5/1) = 5/20 = 1/4. Great! We’ve simplified the constants. Now let’s move on to the 'p' terms. We have p^(4/3) * p^3 in the numerator and p^(2/3) in the denominator. Remember, when multiplying terms with the same base, we add the exponents: p^(4/3) * p^3 = p^((4/3)+3). To add these exponents, we need a common denominator. So, 3 can be written as 9/3. Thus, (4/3) + (9/3) = 13/3. So, p^(4/3) * p^3 = p^(13/3). Now we have p^(13/3) in the numerator and p^(2/3) in the denominator. When dividing terms with the same base, we subtract the exponents: p^(13/3) / p^(2/3) = p^((13/3)-(2/3)) = p^(11/3). Awesome! We've simplified the 'p' terms as well. This step is like putting together the pieces of a puzzle. We've simplified the constants and the variables separately, and now we're combining them to get a final, simplified expression. It's all about breaking down the complex into smaller, manageable parts. So, let’s move on to the final step: putting it all together.

Final Simplification

Okay, guys, we're in the home stretch! We’ve simplified the constants to 1/4 and the 'p' terms to p^(11/3). Now, let's put it all together. Our simplified expression is (1/4) * p^(11/3). And that’s it! We've successfully simplified the original expression. It looked super complicated at first, but by breaking it down step by step and applying the rules of exponents, we made it much simpler. This process highlights the power of methodical simplification. When faced with a complex problem, breaking it down into smaller, manageable steps is key. It's like climbing a mountain – you don't try to climb it in one giant leap. You take it one step at a time, and eventually, you reach the summit. In this case, the summit is the simplified expression. So, remember this approach whenever you encounter a challenging mathematical problem. Break it down, simplify each part, and then put it all together. You'll be surprised at how much you can accomplish. And remember, practice makes perfect. The more you work with these types of expressions, the more comfortable and confident you'll become. So, keep practicing, and you'll be a math whiz in no time! Let's recap the entire process to make sure we've got it down pat.

Recapping the Steps

Let’s recap the steps we took to simplify the expression. First, we translated the expression into standard mathematical notation. This was crucial because the original form was a bit ambiguous. We identified the keywords and symbols and rewrote the expression in a way that made it easier to understand. Then, we broke down the expression into its components: the constants and the 'p' terms with their exponents. This helped us focus on each part individually and understand how they fit together. Next, we applied the exponent rules. We used the rule (a^m)n = a^(m*n) to simplify the powers of 'p'. This significantly reduced the complexity of the expression. After that, we combined like terms. We simplified the constants by performing the multiplication and division. Then, we simplified the 'p' terms by adding and subtracting the exponents, using the rules for multiplying and dividing terms with the same base. Finally, we put it all together to get the simplified expression: (1/4) * p^(11/3). By recapping these steps, we can see the logical flow of the simplification process. Each step builds upon the previous one, leading us closer to the final answer. This structured approach is not just useful for simplifying algebraic expressions but also for tackling any complex problem in mathematics or in life. So, remember these steps, and you'll be well-equipped to handle whatever challenges come your way. Now, let’s address the original question and provide a concise answer.

Final Answer

So, after all that work, the simplified form of the expression seperlima p ^ 2/3^2 * 1/4 p ^ 3/2 ^ 2 / 1/5 P pangkat sepertiga pangkat 2 is (1/4) * p^(11/3). We took a complicated-looking expression and turned it into something much cleaner and easier to understand. Remember, the key is to break things down, apply the rules, and take it step by step. You got this! And that wraps up our journey through simplifying this algebraic expression. I hope you found this guide helpful and that you're feeling more confident about tackling similar problems. Keep practicing, and you'll become a pro in no time! Remember, math is not about memorizing formulas; it's about understanding the concepts and applying them logically. So, keep exploring, keep questioning, and keep simplifying! You've got the power to unlock the secrets of mathematics, guys. And with that, I'll sign off. Happy simplifying!