Solving 27a³ ÷ 3a¹² × A³b⁵ A Step-by-Step Math Guide

Hey guys! Let's break down this math problem together: 27a³ ÷ 3a¹² × a³b⁵. Math can seem intimidating, but trust me, with a step-by-step approach, it becomes super manageable. We're going to dive deep into each part of this equation, making sure you understand not just the how, but also the why behind each step. So, grab your pencils, and let's get started!

Understanding the Basics

Before we even think about tackling this equation, let's nail down the fundamental principles at play here. We're dealing with exponents, coefficients, and variables – all key players in the world of algebra.

• Exponents are those little numbers chilling up in the air, like the ³ in a³. They tell you how many times to multiply the base (in this case, a) by itself. So, a³ means a * a * a. Understanding exponents is crucial because they dictate how terms combine and simplify in algebraic expressions. The rules of exponents, such as the product rule (a^m * a^n = a^(m+n)) and the quotient rule (a^m / a^n = a^(m-n)), are going to be our best friends in solving this problem.
• Coefficients are the numerical part of a term, like the 27 in 27a³. They're basically the multipliers hanging out in front of the variables. When we perform operations like division or multiplication, we need to pay close attention to these coefficients. They behave just like regular numbers, so we can divide, multiply, add, or subtract them as needed.
• Variables, like a and b in our equation, are the unknowns. They represent values that we might not know yet, but we can still manipulate them using algebraic rules. Variables allow us to express relationships and solve for unknown quantities. In this problem, we have a and b, and we'll need to keep track of how they interact with each other and the exponents.

To really master algebra, it’s not enough just to memorize the rules; you've got to understand why they work. Think of exponents as a shorthand for repeated multiplication. When you multiply terms with the same base, you're essentially adding the number of times that base is multiplied by itself. This is the intuition behind the product rule of exponents. Similarly, when you divide, you're canceling out factors, which leads to the subtraction of exponents in the quotient rule. Grasping these concepts will make solving complex problems like this one feel much more intuitive.

Step 1: Rewriting the Expression

Okay, so our original problem is 27a³ ÷ 3a¹² × a³b⁵. The first thing we need to do is rewrite this expression in a way that's easier to work with. Remember, division is just the inverse of multiplication. So, instead of using the division symbol (÷), we can represent it as a fraction. This makes it clearer how the terms are related and sets us up for simplifying.

Instead of 27a³ ÷ 3a¹², we're going to write it as 27a³ / 3a¹². Now our expression looks like this: (27a³) / (3a¹²) × a³b⁵. See how much cleaner that looks already? By converting the division into a fraction, we've transformed the problem into a form where we can easily apply the rules of algebra. This is a common trick in math – rewriting expressions in equivalent forms to make them simpler to manipulate.

This step is crucial because it changes the way we see the problem. The division symbol can sometimes feel like a barrier, making it harder to visualize the relationships between terms. But when we write it as a fraction, we create a sense of unity between the numerator and the denominator. We can now think about canceling out common factors, simplifying fractions, and applying exponent rules more directly. It's like putting the problem into a language that our algebraic brains understand better. Moreover, this fractional representation makes it easier to spot opportunities for simplification in later steps. We can clearly see which terms are in the numerator and which are in the denominator, making it easier to apply rules of exponents and division.

Step 2: Simplifying the Coefficients

The next step in solving (27a³) / (3a¹²) × a³b⁵ is to simplify the coefficients. Remember, coefficients are just the numerical parts of our terms. In this case, we have 27 in the numerator and 3 in the denominator. Simplifying coefficients is like doing regular arithmetic – we're just dividing one number by another.

So, we have 27 / 3. What's that equal? That's right, it's 9! So, we can replace 27 / 3 with 9 in our expression. This makes our expression look like this: (9a³) / (a¹²) × a³b⁵. See how much simpler things are getting? By tackling the coefficients first, we've already reduced the complexity of the problem.

Simplifying coefficients is a fundamental part of algebraic manipulation. It's like clearing away the clutter before we get to the more complex stuff. When coefficients are simplified, it's easier to see the relationships between the variables and exponents. This simplification process not only makes the numbers smaller and more manageable but also allows us to focus on the algebraic part of the expression. This is a classic strategy in problem-solving: break down a complex problem into smaller, more manageable parts. By dealing with the numerical coefficients first, we’re setting ourselves up for success in the next steps.

Step 3: Applying the Quotient Rule

Alright, now we're getting to the good stuff – applying the quotient rule! Our expression is currently (9a³) / (a¹²) × a³b⁵. The quotient rule comes into play when we're dividing terms with the same base but different exponents. Remember, the quotient rule states that a^m / a^n = a^(m-n). This means that when we divide, we subtract the exponents.

In our expression, we have a³ in the numerator and a¹² in the denominator. Both have the same base (a), so we can apply the quotient rule. We subtract the exponents: 3 - 12 = -9. So, a³ / a¹² simplifies to a⁻⁹. Now our expression looks like this: 9a⁻⁹ × a³b⁵. We're making serious progress here, guys!

The quotient rule is a powerful tool in algebra because it allows us to condense expressions and eliminate terms. It's a direct consequence of the way exponents work – division is the inverse operation of multiplication, so when we divide terms with the same base, we're essentially canceling out factors. The quotient rule is not just a formula to memorize; it’s a reflection of the fundamental relationship between multiplication and division. Applying it effectively requires a solid understanding of exponents and how they interact with each other.

Step 4: Applying the Product Rule

Now that we've used the quotient rule, let's move on to the product rule. Our expression is 9a⁻⁹ × a³b⁵. The product rule, as you might recall, states that a^m * a^n = a^(m+n). This means that when we multiply terms with the same base, we add the exponents. It’s like we're combining the powers of a into one term.

Here, we have a⁻⁹ and a³. Both have the base a, so we can add the exponents: -9 + 3 = -6. So, a⁻⁹ × a³ simplifies to a⁻⁶. Now our expression looks like this: 9a⁻⁶b⁵. We're almost there – just a little bit more simplifying to do!

The product rule is the counterpart to the quotient rule, and together, they form the backbone of exponent manipulation. Understanding the product rule allows us to combine like terms and express them in a more concise form. In this step, we're essentially consolidating the a terms, making the expression more streamlined. This simplification process is crucial for understanding the overall behavior of the expression and for any further calculations we might need to perform.

Step 5: Handling Negative Exponents

We're in the home stretch! Our expression is 9a⁻⁶b⁵. Now, it's generally considered good practice to avoid negative exponents in our final answer. They're not wrong, but we can make the expression look cleaner by rewriting them. Remember, a negative exponent means we take the reciprocal of the base raised to the positive exponent. In other words, a⁻ⁿ = 1 / aⁿ.

So, a⁻⁶ is the same as 1 / a⁶. We can rewrite our expression as 9 × (1 / a⁶) × b⁵. This now looks like (9b⁵) / a⁶. Ta-da! We've successfully eliminated the negative exponent and simplified our expression.

Handling negative exponents is a key skill in algebra. It shows that we understand the relationship between positive and negative powers and how they represent repeated multiplication and division. By converting the negative exponent into a positive one in the denominator, we’re essentially shifting the term from the numerator to the denominator. This is a neat trick that makes the expression easier to interpret and work with in further calculations.

Final Answer

And there you have it, guys! We've successfully simplified 27a³ ÷ 3a¹² × a³b⁵. After all the steps, our final answer is (9b⁵) / a⁶. We took a complex-looking expression and broke it down piece by piece, applying the rules of exponents and algebra. Remember, the key is to take it one step at a time and understand the logic behind each move. Math isn't about memorizing formulas; it's about understanding the underlying principles. Keep practicing, and you'll become algebraic wizards in no time!