Solving: (1/3)^2 * 9^(3/2) * 27^(0.5) Calculation

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Solving the Math Problem: (1/3)² * 9^(3/2) * 27^(0.5)

Hey guys! Let's break down this math problem step-by-step so it's super easy to understand. We're going to tackle the calculation of (1/3)² multiplied by 9^(3/2) multiplied by 27^(0.5). It might look intimidating at first, but trust me, it’s totally manageable when we break it down into smaller parts. We'll cover each component, simplify the exponents, and then multiply everything together. Let’s dive in and make math fun!

Understanding the Components

Before we start crunching numbers, let’s look at each part of the equation:

  • (1/3)²: This means we're squaring the fraction 1/3, which is the same as (1/3) * (1/3).
  • 9^(3/2): This is 9 raised to the power of 3/2. Remember, a fractional exponent means we're dealing with both a power and a root. The denominator (2) indicates a square root, and the numerator (3) indicates raising to the power of 3. So, this is the square root of 9, then raised to the power of 3.
  • 27^(0.5): This means 27 raised to the power of 0.5. The exponent 0.5 is the same as 1/2, which represents a square root. So, we're looking for the square root of 27.

Breaking Down (1/3)²

Let's start with the easiest part. When we square a fraction, we simply square both the numerator (the top number) and the denominator (the bottom number). So:

(1/3)² = (1²)/(3²) = 1/9

So, (1/3)² simplifies to 1/9. Easy peasy!

Simplifying 9^(3/2)

Now, let’s tackle 9^(3/2). Remember, this means we need to take the square root of 9 and then raise it to the power of 3. The square root of 9 is 3, because 3 * 3 = 9. Now, we need to raise this result (3) to the power of 3:

3³ = 3 * 3 * 3 = 27

So, 9^(3/2) simplifies to 27. See? We're making progress!

Tackling 27^(0.5)

Finally, let’s deal with 27^(0.5). This is the same as taking the square root of 27. Now, 27 isn't a perfect square (like 9 or 16), but we can simplify it. We need to find a perfect square that is a factor of 27. The perfect square 9 is a factor, since 27 = 9 * 3. So:

√(27) = √(9 * 3) = √9 * √3 = 3√3

So, 27^(0.5) simplifies to 3√3. We've broken down all the parts, and now we just need to put it all together.

Multiplying Everything Together

Now that we've simplified each component, let's multiply them together:

(1/3)² * 9^(3/2) * 27^(0.5) = (1/9) * 27 * 3√3

First, let’s multiply the whole numbers:

(1/9) * 27 = 27/9 = 3

Now we have:

3 * 3√3

Multiply the remaining whole numbers:

3 * 3√3 = 9√3

So, the final result is 9√3.

Step-by-Step Breakdown Recap

Let’s recap the steps we took to solve this problem:

  1. Break down each component:
    • (1/3)²
    • 9^(3/2)
    • 27^(0.5)
  2. Simplify (1/3)²:
    • (1/3)² = 1/9
  3. Simplify 9^(3/2):
    • √9 = 3
    • 3³ = 27
  4. Simplify 27^(0.5):
    • √(27) = √(9 * 3) = 3√3
  5. Multiply all the simplified parts:
    • (1/9) * 27 * 3√3
  6. Multiply (1/9) * 27:
    • 27/9 = 3
  7. Multiply 3 * 3√3:
    • 9√3

So, the final answer is 9√3.

Why Breaking It Down Matters

The key to tackling complex math problems is to break them down into smaller, more manageable steps. When you look at the original equation ((1/3)² * 9^(3/2) * 27^(0.5)), it might seem overwhelming. But by breaking it down, we were able to simplify each part individually and then combine the results. This approach makes the whole process less intimidating and helps prevent errors.

Think of it like building with LEGOs. You start with a big pile of bricks, but you follow the instructions step-by-step, combining smaller pieces to create the final model. Math is similar – each step is like a small piece, and when you put them all together correctly, you get the solution.

Common Mistakes to Avoid

When working with exponents and roots, it's easy to make mistakes if you're not careful. Here are a few common pitfalls to watch out for:

  • Forgetting the order of operations: Always remember PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). In our problem, we dealt with exponents before multiplication.
  • Misunderstanding fractional exponents: A fractional exponent like 3/2 means you're taking both a root and a power. The denominator is the root, and the numerator is the power.
  • Incorrectly simplifying square roots: When simplifying a square root like √27, make sure to factor out the largest perfect square (in this case, 9).
  • Making arithmetic errors: Simple multiplication or division mistakes can throw off the entire calculation. Double-check your work, especially when dealing with multiple steps.

Tips for Mastering Math Calculations

Here are a few tips to help you improve your math skills and tackle complex calculations with confidence:

  • Practice Regularly: Like any skill, math gets easier with practice. Try working through problems every day to reinforce your understanding.
  • Break Down Problems: As we saw in our example, breaking down problems into smaller steps makes them much easier to handle.
  • Understand the Concepts: Don’t just memorize formulas; make sure you understand the underlying concepts. This will help you apply the right techniques in different situations.
  • Check Your Work: Always double-check your calculations to catch any errors.
  • Use Resources: There are tons of great resources available, including textbooks, online tutorials, and math apps. Don’t hesitate to use them!

Real-World Applications

You might be wondering, “When will I ever use this in real life?” Well, understanding exponents and roots is essential in many fields, including:

  • Science: Exponents are used in physics to describe exponential growth and decay, such as radioactive decay or population growth.
  • Engineering: Engineers use exponents and roots in calculations involving stress, strain, and structural integrity.
  • Finance: Compound interest calculations involve exponents, helping you understand how your investments grow over time.
  • Computer Science: Exponents are used in algorithms and data structures, such as binary search trees.

So, while this calculation might seem abstract, the underlying concepts are crucial in many practical applications.

Alternative Methods for Solving

While we solved this problem by simplifying each part and then multiplying, there are other approaches you could take. For example, you could convert all the numbers to the same base (3) before simplifying. Let's try that:

  • (1/3)² = (3⁻¹)² = 3⁻²
  • 9^(3/2) = (3²)^(3/2) = 3^(2 * 3/2) = 3³
  • 27^(0.5) = (3³)^(1/2) = 3^(3/2)

Now, the equation becomes:

3⁻² * 3³ * 3^(3/2)

When multiplying numbers with the same base, you add the exponents:

3^(-2 + 3 + 3/2) = 3^(1 + 3/2) = 3^(5/2)

Now, let's convert this back to radical form:

3^(5/2) = √(3⁵) = √(3⁴ * 3) = 3²√3 = 9√3

We got the same answer! This method demonstrates how different approaches can lead to the same solution, reinforcing the flexibility of mathematical thinking.

Conclusion: Mastering Exponents and Roots

Alright, guys, we’ve successfully calculated (1/3)² * 9^(3/2) * 27^(0.5), and the answer is 9√3. Remember, the key is to break down the problem into smaller, manageable steps, simplify each part, and then combine the results. By understanding the concepts and practicing regularly, you can master exponents and roots and tackle even the trickiest math problems with confidence. Keep practicing, and you’ll be a math whiz in no time!