Simplifying Expressions A Comprehensive Guide

by Marta Kowalska 46 views

Hey guys! Let's dive into the world of algebraic expressions and simplify them like pros. Today, we're tackling the expression: (xy3y2)(z3y2x2)\left(\frac{x y^3}{y^2}\right)\left(\frac{z^3}{y^2 x^2}\right). Simplifying expressions might seem daunting at first, but trust me, it’s like solving a puzzle. We'll break it down step by step so you can understand every twist and turn. Our main goal here is to eliminate variables, tidy up the expression, and make it look as sleek as possible. So, grab your thinking caps, and let’s get started!

Understanding the Basics of Simplifying Expressions

Before we jump into the nitty-gritty, it's crucial to grasp the basic principles of simplifying algebraic expressions. When we talk about simplifying, we mean to reduce an expression to its most basic form. This involves combining like terms, applying exponent rules, and, in our case, eliminating variables where possible.

Key Concepts and Rules

  1. Combining Like Terms: Like terms are those that have the same variables raised to the same powers. For example, 3x23x^2 and βˆ’5x2-5x^2 are like terms, but 3x23x^2 and 3x3x are not. We can combine like terms by adding or subtracting their coefficients. For instance, 3x2+(βˆ’5x2)=βˆ’2x23x^2 + (-5x^2) = -2x^2.

  2. Exponent Rules: Exponents can be a bit tricky, but once you get the hang of the rules, you'll be golden. Here are some essential exponent rules:

    • Product of Powers: When multiplying powers with the same base, add the exponents: amβ‹…an=am+na^m \cdot a^n = a^{m+n}.
    • Quotient of Powers: When dividing powers with the same base, subtract the exponents: aman=amβˆ’n\frac{a^m}{a^n} = a^{m-n}.
    • Power of a Power: When raising a power to another power, multiply the exponents: (am)n=amn(a^m)^n = a^{mn}.
    • Power of a Product: The power of a product is the product of the powers: (ab)n=anbn(ab)^n = a^n b^n.
    • Power of a Quotient: The power of a quotient is the quotient of the powers: (ab)n=anbn\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}.
    • Zero Exponent: Any non-zero number raised to the power of 0 is 1: a0=1a^0 = 1 (where aβ‰ 0a \neq 0).
    • Negative Exponent: A negative exponent means we take the reciprocal of the base raised to the positive exponent: aβˆ’n=1ana^{-n} = \frac{1}{a^n}.
  3. Identity Property of Multiplication: This property states that any number multiplied by 1 remains unchanged. In algebraic terms, aβ‹…1=aa \cdot 1 = a. This is particularly useful when we're trying to eliminate variables by canceling out common factors.

Breaking Down the Given Expression

Now that we've refreshed our memory on the basics, let's dive back into our expression: (xy3y2)(z3y2x2)\left(\frac{x y^3}{y^2}\right)\left(\frac{z^3}{y^2 x^2}\right). To simplify this, we'll first tackle each fraction separately and then combine them.

First, consider the fraction xy3y2\frac{x y^3}{y^2}. We can use the quotient of powers rule to simplify the yy terms. Remember, aman=amβˆ’n\frac{a^m}{a^n} = a^{m-n}. So, y3y2=y3βˆ’2=y1=y\frac{y^3}{y^2} = y^{3-2} = y^1 = y. Thus, the first fraction simplifies to xβ‹…yx \cdot y.

Next, we look at the second fraction z3y2x2\frac{z^3}{y^2 x^2}. There’s not much we can simplify here directly since the variables are different and there are no common factors to cancel out.

Now, let’s combine the simplified first fraction with the second fraction: (xy)(z3y2x2)(x y) \left(\frac{z^3}{y^2 x^2}\right). This gives us xyz3y2x2\frac{x y z^3}{y^2 x^2}.

Step-by-Step Simplification of (xy3y2)(z3y2x2)\left(\frac{x y^3}{y^2}\right)\left(\frac{z^3}{y^2 x^2}\right)

Let’s break down the simplification process into manageable steps. This will help you see how we apply the exponent rules and the identity property of multiplication.

Step 1: Simplify the First Fraction

We start with the first part of the expression: xy3y2\frac{x y^3}{y^2}.

As we discussed earlier, we can simplify the yy terms using the quotient of powers rule: y3y2=y3βˆ’2=y\frac{y^3}{y^2} = y^{3-2} = y. So, this fraction simplifies to xyx y.

Step 2: Rewrite the Expression

Now, let’s rewrite the entire expression using the simplified first fraction:

(xy3y2)(z3y2x2)\left(\frac{x y^3}{y^2}\right)\left(\frac{z^3}{y^2 x^2}\right) becomes (xy)(z3y2x2)(x y) \left(\frac{z^3}{y^2 x^2}\right).

Step 3: Combine the Fractions

Next, we combine these two terms into a single fraction:

(xy)(z3y2x2)=xyz3y2x2(x y) \left(\frac{z^3}{y^2 x^2}\right) = \frac{x y z^3}{y^2 x^2}.

Step 4: Simplify the Combined Fraction

Now, we simplify this fraction by applying the quotient of powers rule to both xx and yy terms:

  • For xx terms: xx2=x1βˆ’2=xβˆ’1=1x\frac{x}{x^2} = x^{1-2} = x^{-1} = \frac{1}{x}.
  • For yy terms: yy2=y1βˆ’2=yβˆ’1=1y\frac{y}{y^2} = y^{1-2} = y^{-1} = \frac{1}{y}.
  • The z3z^3 term remains as it is since there’s no corresponding zz term in the denominator.

So, xyz3y2x2\frac{x y z^3}{y^2 x^2} simplifies to z3xy\frac{z^3}{x y}.

Step 5: Final Simplified Expression

Thus, the simplified expression is z3xy\frac{z^3}{x y}.

Identifying the Eliminated Variable and the Numerator

Now, let’s circle back to the initial questions about the expression.

The Eliminated Variable

The first question asks which variable can be entirely eliminated by applying the identity property of multiplication. Looking at our original expression and the simplification process, we can see that the variable that gets entirely eliminated is x. In the step where we simplified xx2\frac{x}{x^2} to 1x\frac{1}{x}, we effectively canceled out one xx term, showcasing the application of the identity property. So, the variable x can be entirely eliminated in the first fraction simplification.

The Numerator of the Simplified Expression

The second part of the statement asks about the numerator of the simplified expression. After simplifying (xy3y2)(z3y2x2)\left(\frac{x y^3}{y^2}\right)\left(\frac{z^3}{y^2 x^2}\right), we arrived at z3xy\frac{z^3}{x y}. So, the numerator of the simplified expression is z3z^3. It's super clear that z3z^3 sits proudly at the top of our fraction after all the simplification magic.

Common Mistakes to Avoid When Simplifying Expressions

Simplifying expressions can sometimes feel like navigating a maze, and it's easy to stumble into common pitfalls. Let’s highlight some of these mistakes so you can steer clear of them.

1. Incorrectly Applying Exponent Rules

One of the most frequent errors is misapplying exponent rules. Remember, when you’re multiplying powers with the same base, you add the exponents, not multiply them. For instance, x2β‹…x3=x2+3=x5x^2 \cdot x^3 = x^{2+3} = x^5, not x6x^6. Similarly, when dividing powers, you subtract the exponents. Watch out for these subtle yet crucial distinctions.

2. Combining Unlike Terms

Another common mistake is combining terms that aren’t alike. You can only add or subtract terms that have the same variable raised to the same power. For example, you can combine 3x23x^2 and βˆ’5x2-5x^2 to get βˆ’2x2-2x^2, but you can’t combine 3x23x^2 and 3x3x because they are not like terms.

3. Forgetting the Order of Operations

The order of operations (PEMDAS/BODMAS) is your best friend in math. Always remember to perform operations in the correct order: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Skipping this order can lead to incorrect simplifications.

4. Neglecting Negative Signs

Negative signs can be sneaky devils. Make sure to distribute negative signs correctly, especially when dealing with expressions inside parentheses. For example, βˆ’(xβˆ’2)-(x - 2) should be distributed as βˆ’x+2-x + 2, not βˆ’xβˆ’2-x - 2.

5. Overcomplicating Simplification

Sometimes, the simplest approach is the best. Avoid overcomplicating the simplification process by trying to do too much at once. Break down the expression into smaller, manageable parts, simplify each part, and then combine them. This step-by-step approach reduces the chance of errors.

6. Misunderstanding the Identity Property

The identity property of multiplication is powerful but can be misused. Remember, it states that any number multiplied by 1 remains unchanged. When canceling out terms, ensure you are effectively multiplying by 1. For example, xx=1\frac{x}{x} = 1, and this allows you to simplify expressions correctly.

Practice Problems for Mastering Simplification

To truly master simplifying algebraic expressions, practice is key. Here are a few problems for you to try. Work through them step-by-step, and don’t forget to apply the rules and tips we’ve discussed.

Practice Problem 1

Simplify: (a4b2c3)(c5a2b)\left(\frac{a^4 b^2}{c^3}\right)\left(\frac{c^5}{a^2 b}\right)

Practice Problem 2

Simplify: (2x3y2)24x4y3\frac{(2x^3 y^2)^2}{4x^4 y^3}

Practice Problem 3

Simplify: (p3qβˆ’2r4)βˆ’1\left(\frac{p^3 q^{-2}}{r^4}\right)^{-1}

Solutions

  1. a2bc21\frac{a^2 b c^2}{1} or a2bc2a^2bc^2
  2. x2yx^2y
  3. q2r4p3\frac{q^2r^4}{p^3}

Conclusion: Becoming a Simplification Superstar

Simplifying algebraic expressions might seem like a Herculean task at first, but with a solid grasp of the basics and a healthy dose of practice, you'll become a simplification superstar in no time. Remember, the key is to break down the expression, apply the rules methodically, and avoid common pitfalls. Keep practicing, and you’ll find that simplifying expressions becomes second nature. You've got this, guys! Keep up the awesome work, and happy simplifying!