Solving Algebraic Expressions 10 (a² B³ ) ⁴ X (10 B²) ³ A Step-by-Step Guide

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Hey guys! Today, we're going to embark on an exciting journey into the world of algebra. We'll be dissecting a seemingly complex expression, 10 (a² b³ ) ⁴ x (10 b²) ³, and breaking it down step-by-step so you can conquer similar challenges with confidence. Whether you're a student grappling with exponents or just a curious mind eager to expand your mathematical horizons, this article is for you. So, buckle up and let's dive in!

Demystifying the Expression: 10 (a² b³ ) ⁴ x (10 b²) ³

At first glance, the expression 10 (a² b³ ) ⁴ x (10 b²) ³ might seem like a jumble of numbers, variables, and exponents. But don't worry, we're going to unravel it piece by piece. The key to tackling algebraic expressions lies in understanding the order of operations and the rules that govern exponents. Let's start by identifying the different components of our expression.

  • Constants: These are the numerical values in the expression, in our case, the number 10.
  • Variables: These are the letters representing unknown quantities, such as 'a' and 'b' in our expression.
  • Exponents: These are the small numbers written above and to the right of a number or variable, indicating the power to which it is raised. For instance, in a², the exponent is 2, meaning 'a' is multiplied by itself.
  • Parentheses: These are used to group terms together, and operations within parentheses are performed before those outside.
  • Multiplication: Represented by the 'x' symbol, multiplication is one of the fundamental operations we'll be dealing with.

Now that we've identified the key players, let's talk strategy. The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), will be our guiding principle. This tells us the sequence in which we should perform the operations to arrive at the correct answer. Additionally, we'll need to recall the rules of exponents, such as the power of a product rule and the power of a power rule.

Step-by-Step Breakdown: Unleashing the Power of Exponents

Our primary focus in simplifying 10 (a² b³ ) ⁴ x (10 b²) ³ will be on handling the exponents. Exponents, those little numbers perched atop variables and constants, hold immense power in mathematical expressions. Understanding how they work is crucial for simplifying complex equations and unlocking their hidden potential. So, let's delve into the fascinating world of exponents and see how they impact our expression.

First, we encounter the term (a² b³ ) ⁴. This is where the power of a product rule comes into play. This rule states that when raising a product to a power, you must raise each factor within the product to that power. In simpler terms, it means we need to apply the exponent 4 to both a² and b³. This transformation is a fundamental step in simplifying expressions, and it sets the stage for further calculations.

Applying the power of a product rule to (a² b³ ) ⁴, we get a^(24) b^(34), which simplifies to a⁸ b¹². This seemingly simple step has profound implications, as it transforms the complex term into a more manageable form. By distributing the exponent, we've effectively broken down the expression into its constituent parts, making it easier to work with. This is a common theme in algebra: breaking down complex problems into smaller, more manageable steps.

Next, we encounter the term (10 b²) ³. Here, we again apply the power of a product rule. We raise both 10 and b² to the power of 3. This gives us 10³ and b^(2*3), which simplifies to 1000 and b⁶ respectively. Again, this step highlights the importance of the power of a product rule in simplifying algebraic expressions. By applying this rule, we've successfully transformed a complex term into a more understandable form.

By understanding and applying the power of a product rule, we've made significant progress in simplifying our expression. We've successfully dealt with the exponents within the parentheses, and we're now ready to move on to the next stage of our journey. This step-by-step approach is the key to conquering complex algebraic expressions, and it's a skill that will serve you well in your mathematical endeavors.

The Power of a Power Rule: Simplifying Exponents Further

Now, let's talk about the power of a power rule. This rule states that when you raise a power to another power, you multiply the exponents. This is precisely what we did in the previous step! When we had (a²)⁴, we multiplied the exponents 2 and 4 to get a⁸. Similarly, when we had (b³ ) ⁴, we multiplied 3 and 4 to get b¹². This rule is essential for simplifying expressions with nested exponents, and it's a cornerstone of algebraic manipulation. Guys, mastering this rule will significantly boost your algebra skills.

This might seem like a small detail, but it's a crucial element of simplifying algebraic expressions. The power of a power rule allows us to condense complex exponential expressions into simpler forms, making them easier to understand and manipulate. It's a powerful tool in our algebraic arsenal, and it's one that we'll use frequently in our journey to conquer this expression.

Multiplication Magic: Combining Like Terms

With the exponents handled, we can now turn our attention to the multiplication part of the expression 10 (a² b³ ) ⁴ x (10 b²) ³. Remember, we've already simplified (a² b³ ) ⁴ to a⁸ b¹² and (10 b²) ³ to 1000 b⁶. So, our expression now looks like this: 10 * a⁸ b¹² * 1000 b⁶.

The beauty of algebra lies in its ability to combine like terms. In our case, we can multiply the constants (10 and 1000) and combine the 'b' terms. Multiplying 10 and 1000 gives us 10000. Now, let's tackle the 'b' terms. We have b¹² and b⁶. When multiplying terms with the same base, you add the exponents. So, b¹² * b⁶ = b^(12+6) = b¹⁸.

This process of combining like terms is a fundamental aspect of simplifying algebraic expressions. It allows us to consolidate multiple terms into a single, more concise term, making the expression easier to read and understand. This is a critical skill for any aspiring mathematician, and it's one that will serve you well in various mathematical contexts.

The Grand Finale: Presenting the Simplified Expression

After all the hard work, we've finally arrived at the simplified form of our expression! Combining the results from the previous steps, we get 10000 a⁸ b¹⁸. This is the simplified version of the original expression 10 (a² b³ ) ⁴ x (10 b²) ³. Phew! We made it! You see, guys, even the most intimidating expressions can be tamed with a systematic approach and a solid understanding of the rules.

The simplified expression, 10000 a⁸ b¹⁸, is much easier to understand and work with than the original. It clearly shows the relationship between the constants and the variables, and it highlights the power of exponents in shaping the expression's behavior. This is the ultimate goal of simplification: to reveal the underlying structure and meaning of an expression.

Real-World Applications: Why Does This Matter?

You might be wondering,