Problem 83 - Applied Calculus

Question

Simplify. $$ \frac{\left(9 x y^{3} z\right)^{2}}{3(x y z)^{2}} $$

Step-by-Step Solution

Answer

The simplified expression is \(27y^4\).

Step 1: Simplify the Numerator

First, simplify the expression in the numerator: \( (9xy^3z)^2 \). This means multiplying each term inside the parentheses by itself:\[(9xy^3z)^2 = 9^2 \, (xy^3z)^2 = 81x^2y^6z^2\]

Step 2: Simplify the Denominator

Now simplify the expression in the denominator: \( 3(xyz)^2 \). You need to square each term inside the parentheses and multiply by 3:\[3(xyz)^2 = 3 \, (x^2y^2z^2) = 3x^2y^2z^2\]

Step 3: Divide the Numerator by the Denominator

Next, divide each term in the numerator by the corresponding term in the denominator:\[\frac{81x^2y^6z^2}{3x^2y^2z^2} = \frac{81}{3} \cdot \frac{x^2}{x^2} \cdot \frac{y^6}{y^2} \cdot \frac{z^2}{z^2}\]Simplifying each part yields:\[27 \cdot 1 \cdot y^4 \cdot 1 = 27y^4\]

Step 4: Final Answer

Upon simplifying the expression, the final result is:\[27y^4\]

Key Concepts

Simplifying Fractions in Algebra

In algebra, simplifying fractions means reducing them to their simplest form. This is much like simplifying numerical fractions, but with algebraic expressions instead. To simplify an algebraic fraction, the goal is to cancel out common factors in the numerator and the denominator.

When you have a fraction where both the numerator and the denominator are expressions, follow these steps:

• Factor both the numerator and the denominator if possible.
• Cancel out any factors that appear in both the top and bottom of the fraction.

In the given exercise, you're simplifying an algebraic fraction. The process involves first dealing with multiplication and then reducing the fraction by dividing out common factors. This ensures that you end up with the most simplified form of the expression. Remember that the power of algebra comes from making expressions simpler and more manageable. Simplifying is a key part of that process.

Exponents and Powers

Understanding exponents and powers is crucial in algebra. An exponent tells us how many times a number, known as the base, is multiplied by itself. For instance, in the expression \[9^2\] nine is the base and 2 is the exponent. This means you'll multiply 9 by itself, resulting in 81.

Here's what you need to know:

• If you see an expression like \[(xy^3z)^2\], you'll apply the exponent to each term inside the parentheses, resulting in \[x^2y^6z^2\].
• This rule is a cornerstone in reducing expressions involving exponents, and it's quite handy when simplifying algebraic fractions, as seen in the exercise.

Exponents allow us to express what might be complex multiplication in a very compact form, saving space and helping us see patterns more clearly. Mastering this will make simplifying expressions a lot simpler.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions using basic algebraic rules. This process often involves combining like terms, factoring, expanding expressions, and applying reliable strategies like the distributive law.

A few essential guidelines for algebraic manipulation include:

• The order of operations - remember the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
• Maintaining equality while manipulating both sides of an equation.
• Always double-check your work to ensure each step is mathematically valid.

In our exercise, you manipulate the expression by simplifying both the numerator and denominator separately. Then, by dividing the simplified forms, you break down complex expressions into simpler ones to derive the final answer. Algebraic manipulation gives us the tools to solve and simplify expressions effectively, making it crucial for mastering algebra.