Simplify Algebraic Expressions: A Step-by-Step Guide

Hey everyone! Today, we're diving deep into the awesome world of simplifying algebraic expressions. It might sound a bit intimidating at first, but trust me, guys, once you get the hang of it, it's actually super satisfying and even kinda fun! We'll be tackling a specific problem today: simplifying the expression $\frac{4 y z^2}{3 x y^4 z} \cdot \frac{15 x^2 y}{8 x^3 y z}$. This might look like a jumbled mess of letters and numbers, but by breaking it down piece by piece, we'll uncover its hidden, simplified form. So, grab your thinking caps, maybe a snack, and let's get this done!

Understanding the Basics: What Does "Simplify" Even Mean?

So, what exactly does it mean to simplify an algebraic expression, especially when you're multiplying fractions like the one we've got? In simple terms, simplifying means rewriting an expression in its most basic, compact form. Think of it like tidying up your room; you're taking all the scattered items (numbers, variables, exponents) and putting them in their proper places so everything is neat and easy to understand. For algebraic expressions, this usually involves combining like terms, canceling out common factors in the numerator and denominator, and reducing fractions to their lowest terms. When we're dealing with fractions, especially those with variables, simplification often means we can cancel out parts of the expression that appear in both the top (numerator) and the bottom (denominator). This is like finding pairs of socks that match and putting them away – they're redundant and can be removed! The goal is to end up with an expression that is equivalent to the original but much easier to work with. This is crucial in mathematics because a simplified expression is easier to evaluate, compare, and use in further calculations. Imagine trying to solve a complex equation with a super long, messy expression – it would be a nightmare! Simplification is the key that unlocks easier problem-solving.

Tackling the Problem: Step-by-Step Simplification

Alright, let's get down to business with our specific problem: $\frac{4 y z^2}{3 x y^4 z} \cdot \frac{15 x^2 y}{8 x^3 y z}$. The first thing we notice is that we're multiplying two fractions. To multiply fractions, we multiply the numerators together and the denominators together. So, our expression becomes:

$$\frac{(4 y z^2) \cdot (15 x^2 y)}{(3 x y^4 z) \cdot (8 x^3 y z)}$$

Now, let's simplify the numerator and the denominator separately. First, the numerator: $(4 y z^2) \cdot (15 x^2 y)$. We multiply the numerical coefficients (4 and 15) and then combine the variables. Remember that when you multiply variables with exponents, you add the exponents. So, $y \cdot y$ is $y^{1+1} = y^2$. We have:

$$4 \cdot 15 = 60$$

$$y \cdot y = y^2$$

$$z^2$$

$$x^2$$

Putting it all together, the numerator is $60 x^2 y^2 z^2$.

Now, let's look at the denominator: $(3 x y^4 z) \cdot (8 x^3 y z)$. Again, we multiply the coefficients and combine the variables:

$$3 \cdot 8 = 24$$

$$x \cdot x^3 = x^{1+3} = x^4$$

$$y^4 \cdot y = y^{4+1} = y^5$$

$$z \cdot z = z^{1+1} = z^2$$

So, the denominator is $24 x^4 y^5 z^2$.

Now, our expression looks like this:

$$\frac{60 x^2 y^2 z^2}{24 x^4 y^5 z^2}$$

This is where the simplification really kicks in! We need to reduce this fraction by canceling out any common factors between the numerator and the denominator. Let's start with the numerical coefficients: 60 and 24. The greatest common divisor (GCD) of 60 and 24 is 12. So, we divide both by 12:

$$\frac{60}{12} = 5$$

$$\frac{24}{12} = 2$$

So the numerical part becomes $\frac{5}{2}$.

Now, let's handle the variables. Remember the rule for dividing exponents: $\frac{a^m}{a^n} = a^{m-n}$.

For $x$: $\frac{x^2}{x^4} = x^{2-4} = x^{-2}$. This means $x^2$ in the numerator and $x^4$ in the denominator leave us with $x^2$ in the denominator (since $x^{-2}$ is the same as $\frac{1}{x^2}$).

For $y$: $\frac{y^2}{y^5} = y^{2-5} = y^{-3}$. This means $y^2$ in the numerator and $y^5$ in the denominator leave us with $y^3$ in the denominator.

For $z$: $\frac{z^2}{z^2} = z^{2-2} = z^0 = 1$. The $z^2$ terms cancel out completely!

Putting it all together, we have:

$$\frac{5}{2} \cdot \frac{1}{x^2} \cdot \frac{1}{y^3} \cdot 1 = \frac{5}{2 x^2 y^3}$$

And there you have it! The simplified expression is $\frac{5}{2 x^2 y^3}$. This is so much cleaner than the original, right?

Breaking Down the Choices: Finding the Correct Answer

Now that we've done the hard work of simplifying the expression, let's look at the options provided to see which one matches our result. Remember, we simplified $\frac{4 y z^2}{3 x y^4 z} \cdot \frac{15 x^2 y}{8 x^3 y z}$ to $\frac{5}{2 x^2 y^3}$. Let's check each option:

• A. $\frac{2 x^2 y^3}{5}$: This doesn't match our result. The variables are in the wrong place, and the numbers are inverted.
• B. $\frac{2 x^2 y^3}{5 z}$: This also doesn't match. Again, the variables are in the numerator, and there's an extra 'z' term that shouldn't be there.
• C. $\frac{5 z}{2 x^2 y^3}$: This is close, but our $z$ terms canceled out completely, so there shouldn't be a $z$ in the numerator.
• D. $\frac{5}{2 x^2 y^3}$: Bingo! This perfectly matches the result we got after careful simplification. The numerical coefficients are correct ($\frac{5}{2}$), and the variables ($x^2 y^3$) are correctly placed in the denominator.

So, the correct answer is D. It’s always a good idea to double-check your work, especially when dealing with exponents and fractions, as small mistakes can easily happen. But by following the rules systematically, we can confidently arrive at the correct simplified form.

Why Simplifying Expressions Matters in Math

So, why do we even bother with all this simplifying algebraic expressions stuff? It's not just about making problems look neater; it's a fundamental skill that underpins almost everything you'll do in higher-level mathematics. Think about solving equations. If you have a super complicated equation, the first step is almost always to simplify both sides to make it manageable. If you're working with functions, simplifying their expressions can reveal key properties, like asymptotes or intercepts, that would be hidden in a more complex form. In calculus, simplifying expressions is essential for finding derivatives and integrals more easily. Imagine trying to differentiate $\frac{4 y z^2}{3 x y^4 z} \cdot \frac{15 x^2 y}{8 x^3 y z}$ in its original form versus differentiating $\frac{5}{2 x^2 y^3}$ – the difference is staggering! Furthermore, understanding simplification helps build intuition about how mathematical objects relate to each other. It teaches you about the properties of numbers and variables, like the rules of exponents and how to manipulate fractions. It's like learning the grammar of mathematics; once you master it, you can express complex ideas clearly and efficiently. This skill is not just confined to math class either. In fields like physics, engineering, computer science, and economics, mathematical models are used extensively, and the ability to simplify and manipulate these models is crucial for analysis and problem-solving. So, the next time you're simplifying an expression, remember you're not just doing homework; you're building a powerful toolset for understanding and shaping the world around you. It’s all about making complex things understandable, and that's a superpower in any field!

Final Thoughts: Practice Makes Perfect!

Alright guys, that was a solid dive into simplifying algebraic expressions. We took a complicated-looking multiplication of fractions and, step by step, broke it down into its simplest form, $\frac{5}{2 x^2 y^3}$. We reinforced the rules of multiplying and dividing exponents, combined numerical coefficients, and canceled out common factors. Remember, the key takeaways are:

1. Multiply Numerators and Denominators: When multiplying fractions, multiply the top parts together and the bottom parts together.
2. Combine Like Terms: Multiply coefficients and add exponents for the same variables.
3. Cancel Common Factors: Look for identical terms in the numerator and denominator and eliminate them.
4. Apply Exponent Rules: Use $\frac{a^m}{a^n} = a^{m-n}$ for division and $a^m \cdot a^n = a^{m+n}$ for multiplication.

Simplifying expressions is a fundamental skill that gets easier with practice. Don't be afraid to tackle more problems! The more you practice, the more comfortable you'll become with the rules and the faster you'll be able to spot opportunities for simplification. Keep experimenting, keep learning, and you'll become a pro at this in no time. Happy simplifying!