Problem 75 - Brief 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 squaring each term inside the parentheses:\( (9xy^3z)^2 = 9^2 imes x^2 imes (y^3)^2 imes z^2 \).Calculating each square, we get \( 9^2 = 81 \), \( (y^3)^2 = y^{6} \), so the numerator becomes \( 81x^2y^6z^2 \).

Step 2: Simplify the Denominator

Now, simplify the expression in the denominator \( 3(xyz)^2 \). We square each term inside the parentheses:\( 3(xyz)^2 = 3 imes x^2 imes y^2 imes z^2 \).

Step 3: Divide Terms

Next, divide each term in the numerator by the corresponding term in the denominator:\( \frac{81x^2y^6z^2}{3x^2y^2z^2} \).Divide the coefficients: \( \frac{81}{3} = 27 \).Divide the variables:- For \( x \): \( \frac{x^2}{x^2} = 1 \)- For \( y \): \( \frac{y^6}{y^2} = y^{4} \)- For \( z \): \( \frac{z^2}{z^2} = 1 \).

Step 4: Combine and Simplify

Combine the results from the division: The simplified expression is \( 27y^4 \).

Key Concepts

Exponent Rules

Understanding exponent rules is fundamental when dealing with algebraic simplification, particularly in expressions involving powers. Exponents indicate how many times a number, known as the base, is multiplied by itself. For instance, in the term \( y^3 \), "\( y \)" is the base, and "3" is the exponent, signalling that \( y \) is multiplied by itself three times: \( y \times y \times y \).

When simplifying expressions with exponents, remember these key rules:

• **Product of Powers Rule:** When multiplying like bases, add their exponents. For example, \( a^m \times a^n = a^{m+n} \).
• **Power of a Power Rule:** When raising a power to another power, multiply the exponents: \( (a^m)^n = a^{m\times n} \).
• **Power of a Product Rule:** To raise a product to a power, raise each factor to the power separately: \((ab)^n = a^n \times b^n \).
• **Quotient of Powers Rule:** When dividing like bases, subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n} \).

In our specific problem, we first apply the Power of a Product Rule. This involves squaring each component in the numerator: \((9xy^3z)^2\) becomes \(9^2 \times x^2 \times (y^3)^2 \times z^2\). Using the Power of a Power Rule, \((y^3)^2\) simplifies to \(y^6\). Understanding these rules lets us manage and simplify complex algebraic expressions efficiently.

Fraction Division

Dividing fractions in algebra involves dividing terms component-wise. In a fraction \( \frac{numerator}{denominator} \), you're essentially performing division for each part separately. Let's break it down:

1. **Divide Coefficients:** Handle the numbers first. If you have \( \frac{81}{3} \), simplify it by performing the division to get 27.2. **Divide Variables:** Apply the Quotient of Powers Rule for variables. When you see \( \frac{x^2}{x^2} \), this becomes 1 because \( x^{2-2} = x^0 = 1 \). Similarly, \( \frac{y^6}{y^2} \) becomes \( y^{4} \), as you subtract exponents (\( 6 - 2 = 4 \)). For \( \frac{z^2}{z^2} \), this again simplifies to 1.

To simplify the division of fractions, focus on reducing each component separately. This often transforms a complex algebraic fraction into a much simpler form, helping you to solve and understand problems more intuitively.

Polynomial Expressions

Polynomial expressions are algebraic expressions that consist of variables and coefficients, featuring operations like addition, subtraction, multiplication, and non-negative integer exponents. A polynomial is expressed in terms of its degree, relevant to the highest power of the variable in the expression. For example, \( 3x^4 + 2x^3 + x - 5 \) is a polynomial of degree 4.

Key components of polynomials include:

• **Terms:** Individual parts separated by addition or subtraction signs. In \( 3x^4 + 2x^3 \), "3x^4" and "2x^3" are separate terms.
• **Coefficients:** The numerical part of the terms. In "3x^4", 3 is the coefficient.
• **Degrees:** The exponent of the variable within a term. "4" in "3x^4" denotes the degree of that term.

Polynomial expressions simplify following similar algebraic rules, such as combining like terms and applying exponent rules. When working with polynomials in fraction, you often see terms simplified by division as with any complex fraction.

In our given problem, the polynomial in the simplified numerator was reduced by applying exponent rules and fraction division, showing the polynomial term \( 27y^4 \).