Simplifying Algebraic Expressions: A Step-by-Step Guide

Hey math enthusiasts! Today, we're diving into the exciting world of simplifying algebraic expressions. Specifically, we're going to break down how to solve a problem like this: Which expression is equivalent to $\frac{(5 a b)^3}{30 a^{-6} b^{-7}}$? Assuming that $a$ and $b$ are not equal to zero. This is a common type of question that you might encounter in your algebra studies, and understanding the steps involved will give you a major advantage. Let's get started!

Understanding the Problem: The Foundation of Simplification

Before we jump into the calculations, let's make sure we're all on the same page. The problem asks us to find an expression that is equal to the given expression, but in a simpler form. Think of it like this: we're looking for an equivalent expression that is easier to read and work with. The expression we're starting with involves exponents, coefficients, and variables ($a$ and $b$). The key here is to use our knowledge of exponent rules and algebraic manipulation to get to a simplified version. The expression contains a fraction, with the numerator being $(5ab)^3$ and the denominator being $30a^{-6}b^{-7}$. To simplify, we need to address the exponents, the coefficients, and the variables with their respective powers.

So, why is this important? Well, simplifying expressions is a fundamental skill in algebra. It helps us solve equations, understand relationships between variables, and work with more complex problems down the line. It's like building a house – you need a solid foundation before you can add the walls and the roof. Similarly, a strong grasp of simplification rules will make all other algebra concepts much easier to understand. The ability to manipulate and simplify expressions is critical for solving equations, graphing functions, and working with more advanced mathematical concepts. Being able to break down a complex expression into a simpler form allows for more efficient calculations and a clearer understanding of the underlying relationships between variables and constants. This skill also enhances problem-solving abilities in other areas, as it teaches a methodical approach to tackling complex challenges. Remember, practice makes perfect! The more you work with these types of problems, the more comfortable and confident you'll become. And trust me, it's a very satisfying feeling to simplify a complex expression into something elegant and concise.

Remember the fundamental rules of exponents. The power of a product rule is crucial here: $(xy)^n = x^n y^n$. Also, know how to handle negative exponents: x^{-n} = rac{1}{x^n}. These rules are the bread and butter of our simplification process. And don't forget the quotient rule for exponents: $\frac{x^m}{x^n} = x^{m-n}$. Keeping these rules in mind is like having a toolkit – you've got the right tools to take apart and rebuild the expression in a way that makes it easier to understand. We will use all these rules in the next section.

Step-by-Step Solution: Unveiling the Simplified Expression

Alright, guys, let's get down to the nitty-gritty and simplify that expression step-by-step. We will start with the numerator: $(5ab)^3$. Using the power of a product rule, we raise each term inside the parentheses to the power of 3. So, $(5ab)^3$ becomes $5^3 * a^3 * b^3$. Now, calculate $5^3$, which is $5 * 5 * 5 = 125$. So, the numerator simplifies to $125a^3b^3$. Now we're dealing with $\frac{125a^3b^3}{30a^{-6}b^{-7}}$.

Next, let's tackle the denominator. We have $30a^{-6}b^{-7}$. We'll rewrite this expression to get the variables with positive exponents by moving them to the numerator, and the constant 30, which stays in the denominator. To do this, recall that a negative exponent means the base is in the opposite position in the fraction. So, $a^{-6}$ becomes $a^6$ in the numerator, and $b^{-7}$ becomes $b^7$ in the numerator as well. Now we get $\frac{125a^3b^3 a^6 b^7}{30}$. Now the expression becomes $125a^3b^3$. Then apply the product rule of exponents by combining like terms ($a$ and $b$). Remember, when multiplying terms with the same base, you add the exponents. For the $a$ terms, we have $a^3 * a^6 = a^{3+6} = a^9$. For the $b$ terms, we have $b^3 * b^7 = b^{3+7} = b^{10}$. Combining these, our expression becomes $\frac{125 a^9 b^{10}}{30}$. Finally, we can simplify the fraction $\frac{125}{30}$ by dividing both the numerator and the denominator by their greatest common divisor, which is 5. So, $\frac{125}{5} = 25$ and $\frac{30}{5} = 6$. So, the fully simplified expression is $\frac{25 a^9 b^{10}}{6}$.

Let's recap the steps: First, expand the numerator. Second, adjust the negative exponents. Third, combine like terms using exponent rules. Last, simplify the coefficient fraction. And that's it! We've successfully simplified the expression.

Matching the Solution to the Options: Finding the Correct Answer

Now that we have our simplified expression, $\frac{25 a^9 b^{10}}{6}$, let's see which of the multiple-choice options matches it. We need to compare our result with the answer choices. Remember, the options are:

A. $\frac{a^7 b^{10}}{6}$ B. $\frac{125 a^{18} b^{21}}{30}$ C. $\frac{25 a^3 b^4}{6}$ D. $\frac{25 a^9 b^{10}}{6}$

By comparing our simplified expression, $\frac{25 a^9 b^{10}}{6}$, with the answer choices, we find that option D matches perfectly. The coefficients, and the exponents of the variables $a$ and $b$ all match our simplified result. Thus, the correct answer is D!

Conclusion: Mastering the Art of Simplification

And there you have it, folks! We've successfully simplified the algebraic expression $\frac{(5 a b)^3}{30 a^{-6} b^{-7}}$ and found the equivalent expression. This process involved understanding exponent rules, applying them step-by-step, and careful manipulation of the expression. Remember, practice is key! The more you work through these types of problems, the more confident you'll become in your algebraic abilities. Keep in mind the rules for exponents, the order of operations, and the importance of simplifying fractions. With these tools in your arsenal, you'll be well-equipped to tackle any simplification problem that comes your way. So, keep practicing, and don't be afraid to ask for help if you get stuck. Math can be fun, and with a little effort, you can master these concepts. Good luck, and keep up the great work!