Problem 12 - Technical Mathematics with Calculus

Question

$$\frac{16 y^{2}}{9 x^{2}}-\frac{8 y^{3}}{3 x^{3}}+\frac{24 y^{4}}{9 x}$$

Step-by-Step Solution

Answer

\frac{16 y^{2}}{9 x^{2}} - \frac{8 y^{3}}{3 x^{3}} + \frac{24 y^{4}}{9 x} = \frac{16 y^{2}}{9 x^{2}} - \frac{8 y^{3}}{3 x^{3}} + \frac{8 y^{4}}{3 x}

Step 1: Simplify each term

First, simplify each term individually by reducing fractions. Divide the numerical coefficients and apply the exponent rules for dividing powers with the same base.

Step 2: Combine like terms

After simplification, observe if there are any like terms that can be combined. In this expression, there are no like terms to combine since each term has a different power of y and x.

Step 3: Express final simplified form

Write down the simplified form of each term to express the final simplified expression of the given polynomial.

Key Concepts

Polynomial Simplification

Understanding the concept of polynomial simplification is fundamental in algebra. It involves reducing an expression to its simplest form, making it easier to work with and understand. When faced with a polynomial, look for terms that share the same variables raised to the same power; these are what we call 'like terms.' Simplifying entails combining like terms, which involves adding or subtracting their coefficients.

For instance, if the exercise presents you with terms that can't be combined, like each having different powers of variables, simplification then focuses on reducing each term individually. This brings us to how we handle coefficients and exponents: coefficients can often be divided if they share a common factor, while exponent rules govern how we simplify terms with variables raised to powers. Once each term is simplified, the polynomial as a whole becomes substantially easier to work with, even when the terms cannot be combined any further.

Reducing Fractions

The process of reducing fractions is essentially breaking down a larger fraction into its simplest form, where the numerator and denominator share no common factors other than 1. This not only makes the fraction cleaner and more straightforward, but it also simplifies subsequent calculations you might perform with it.

To reduce a fraction, find the greatest common divisor (GCD) of the numerator and denominator and divide both by this number. In algebraic expressions, reducing fractions also involves simplifying any variables by applying exponent rules. When you're given numerical coefficients in fraction form along with variables, like in the exercise example, you'll approach each term separately. Dividing the numerical parts and then simplifying the variables results in a more manageable expression that still retains its original value.

Exponent Rules

Mastering exponent rules is crucial for simplifying algebraic expressions, especially polynomials. These rules explain how to handle powers of variables when they are multiplied, divided, or raised to another power. Here are some fundamental exponent rules you should know:

• \textbf{Product of Powers:} To multiply terms with the same base, add their exponents: \(a^m \times a^n = a^{m+n}\).
• \textbf{Quotient of Powers:} To divide terms with the same base, subtract the exponent of the divisor from the exponent of the dividend: \(a^m \/ a^n = a^{m-n}\).
• \textbf{Power of a Power:} To take a power of a power, multiply the exponents: \((a^m)^n = a^{mn}\).
• \textbf{Negative Exponent:} A negative exponent indicates a reciprocal: \(a^{-n} = 1\/a^n\).

When simplifying algebraic expressions, apply these rules systematically to each term. For example, when dividing powers with the same base, as seen in the exercise, you subtract the exponents. Following these rules correctly, you'll obtain a simpler expression that can be easily analyzed or further manipulated.