Simplify Expressions With Exponents A Step-by-Step Guide

Hey guys! Ever stumbled upon a math problem that looks like a jumbled mess of exponents and variables? Don't worry, we've all been there. Today, we're going to break down a seemingly complex expression and figure out which of the given options is its true equivalent. Our mission, should we choose to accept it (and we do!), is to simplify $\frac{5 y^3}{(5 y)^{-2}}$. Buckle up, because we're about to dive deep into the world of exponents and algebraic manipulation!

Understanding the Problem

The expression we're tackling is $\frac{5 y^3}{(5 y)^{-2}}$. At first glance, it might seem intimidating, but let's dissect it piece by piece. We've got a fraction, which means we're dealing with division. In the numerator, we have $5y^3$, which means 5 multiplied by $y$ raised to the power of 3. The denominator is where things get a bit trickier: $(5y)^{-2}$. This means the quantity $5y$ is raised to the power of -2. Remember, a negative exponent indicates that we need to take the reciprocal of the base raised to the positive exponent. In simpler terms, $x^{-n} = \frac{1}{x^n}$. This is crucial to understanding how to simplify this expression. So, let's rewrite the denominator using this rule. We get $\frac{1}{(5y)^2}$. Now, our original expression looks like $\frac{5 y^3}{\frac{1}{(5 y)^2}}$. This is a fraction divided by another fraction, which brings us to the next step: dealing with complex fractions.

To handle this, we'll use the rule that dividing by a fraction is the same as multiplying by its reciprocal. This is a fundamental concept in fraction manipulation, and it's super helpful in simplifying expressions like this one. Think of it like flipping the fraction in the denominator and then multiplying. So, instead of dividing by $\frac{1}{(5y)^2}$, we'll multiply by $(5y)^2$. This transforms our expression into a much more manageable form. We now have $5y^3 \cdot (5y)^2$. See? It's already looking less scary! But we're not done yet. We still need to simplify further by applying the power rule for exponents and combining like terms. This involves expanding $(5y)^2$ and then using the rule that $x^m \cdot x^n = x^{m+n}$ to combine the $y$ terms. By breaking down the problem into these smaller, digestible steps, we can systematically work towards the solution. Remember, the key to mastering algebra is to take things one step at a time and understand the underlying rules and principles. Now, let’s move on to the next section and actually perform the simplification!

Step-by-Step Simplification

Okay, let's get our hands dirty and simplify the expression $5y^3 \cdot (5y)^2$. The first thing we need to do is tackle the term $(5y)^2$. Remember the power of a product rule? It states that $(ab)^n = a^n b^n$. In our case, this means $(5y)^2 = 5^2 \cdot y^2$. So, let's calculate $5^2$, which is simply 5 multiplied by itself, giving us 25. Thus, $(5y)^2$ becomes $25y^2$. Now, we can substitute this back into our expression: $5y^3 \cdot 25y^2$. This looks a lot simpler already!

Next, we need to multiply the terms together. We'll start by multiplying the coefficients (the numbers in front of the variables). We have 5 multiplied by 25, which equals 125. So, our expression now looks like $125y^3y^2$. Great! We're almost there. The final step involves dealing with the exponents of the $y$ terms. Remember the product of powers rule? It says that when you multiply terms with the same base, you add their exponents: $x^m \cdot x^n = x^{m+n}$. In our case, we have $y^3$ multiplied by $y^2$. So, we add the exponents 3 and 2, which gives us 5. This means $y^3y^2 = y^5$. Finally, we can substitute this back into our expression, giving us $125y^5$. Voila! We've successfully simplified the expression. It might seem like a lot of steps, but by breaking it down and applying the rules of exponents, we arrived at a clear and concise answer. Now, let's take a look at the original options and see which one matches our simplified expression.

Matching the Result with the Options

We've diligently worked through the simplification process, and we've arrived at the expression $125y^5$. Now, let's compare this result with the options provided: A. $y^5$, B. $y^6$, C. $125y^3$, and D. $125y^5$. It's pretty clear which one matches our result, right? Option D, $125y^5$, is an exact match! This confirms that our simplification was accurate and that we've successfully identified the equivalent expression. Sometimes, math problems try to trick you with similar-looking options, but by carefully applying the rules and breaking down the problem step by step, we can avoid these traps. In this case, options A, B, and C might seem tempting at first glance, but they don't fully account for all the terms and exponents in the original expression. Option A misses the coefficient of 125, while options B and C have the wrong exponent for the $y$ term. This highlights the importance of paying close attention to every detail when simplifying algebraic expressions. A small mistake in one step can lead to a completely different result. So, always double-check your work and make sure you're applying the rules correctly. Now that we've confidently identified the correct answer, let's briefly recap the key steps we took to solve this problem.

Key Concepts and Takeaways

Let's quickly recap the key concepts and takeaways from this problem. We started with the expression $\frac{5 y^3}{(5 y)^{-2}}$ and our goal was to simplify it to an equivalent form. The first crucial step was understanding negative exponents. Remember, $x^{-n} = \frac{1}{x^n}$. This allowed us to rewrite the denominator and get rid of the negative exponent. Then, we dealt with the complex fraction by multiplying by the reciprocal of the denominator. This transformed our division problem into a multiplication problem, making it much easier to handle. Next, we applied the power of a product rule, $(ab)^n = a^n b^n$, to expand $(5y)^2$. This gave us $25y^2$. We then multiplied the coefficients and applied the product of powers rule, $x^m \cdot x^n = x^{m+n}$, to combine the $y$ terms. This led us to our final simplified expression: $125y^5$. Throughout this process, we emphasized the importance of breaking down complex problems into smaller, more manageable steps. This is a valuable strategy not just in math, but in many areas of life. By focusing on one step at a time and applying the relevant rules and principles, we can tackle even the most challenging problems. We also highlighted the significance of paying attention to detail and double-checking our work to avoid errors. A small mistake can sometimes have a big impact on the final result, so it's always a good idea to be thorough and meticulous. Finally, we saw how understanding the fundamental rules of exponents is essential for simplifying algebraic expressions. These rules are like the building blocks of algebra, and mastering them will greatly improve your ability to solve a wide range of problems. So, keep practicing, keep learning, and keep exploring the fascinating world of math!

Which expression is equivalent to the mathematical expression 5y³ / (5y)⁻²?

Simplify Expressions with Exponents A Step-by-Step Guide