Problem 58 - Algebra and Trigonometry

Question

Simplify each exponential expression. $$ \frac{10 x^{4} y^{9}}{30 x^{12} y^{-3}} $$

Step-by-Step Solution

Answer

The simplification of the given expression is \( \frac{y^{12}}{3x^8} \).

Step 1: Simplify the Numerical Coefficients

First, simplify the numerical coefficients, in this case 10 and 30. This is done by performing the division \( \frac{10}{30} \) which simplified is \( \frac{1}{3} \).

Step 2: Simplify the Variable x

Next, simplify the variable \(x\) . This is done by subtracting the exponent in the denominator from the exponent in the numerator, that is \(4 - 12\), which gives \(-8\). Therefore, \(x^{4-12} = x^{-8}\). However, to write an exponential expression in simplified form, the exponent should be positive. Therefore, by the law of exponents, which states that \(a^{-n} = \frac{1}{a^n}\), we have \(x^{-8} = \frac{1}{x^8}\).

Step 3: Simplify the Variable y

Lastly, simplify the variable \(y\). This is done by subtracting the exponent in the denominator from the exponent in the numerator, that is \(9 - -3\), which gives \(12\). Therefore, \(y^{9-(-3)} = y^{12}\).

Key Concepts

Simplifying Fractions

When dealing with fractions, one of the most important steps is simplifying them. This process involves breaking down the fraction into its simplest form. Not only is this important for numerical coefficients, but also when working with variables. To simplify, we reduce both the numerator and the denominator by their greatest common factor (GCF).
For instance, consider the fraction \( \frac{10}{30} \). Both 10 and 30 are divisible by 10, which is their GCF. By dividing the numerator and the denominator by this number, we get \( \frac{1}{3} \), the simplest form of the fraction.
It's crucial to remember that simplifying fractions helps make the expression easier to understand and work with. Even when fractions involve variables, as in \( \frac{x^4 y^9}{x^{12} y^{-3}} \), the same principle applies: simplify each part separately. Always aim for the simplest version, which enhances clarity in mathematical problems.

Laws of Exponents

Understanding the laws of exponents is a key part of working with exponential expressions. These laws help simplify expressions with powers and make them more manageable.
One essential rule is the division of exponents: when dividing like bases, subtract the exponents. For example, in the expression \( \frac{x^4}{x^{12}} \), you subtract the exponent in the denominator from that in the numerator, resulting in \( x^{-8} \). This can often lead to negative exponents.

• Another crucial rule is how to handle negative exponents. The expression \( a^{-n} \) is equivalent to \( \frac{1}{a^n} \). This means we can rewrite \( x^{-8} \) as \( \frac{1}{x^8} \).
• Similarly, handling subtractions with negative exponents can simplify calculations, as seen with \( y^{9-(-3)} = y^{12} \).

These laws form the foundation of simplifying exponential expressions, offering a roadmap to simplified and more understandable outcomes.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and operations. Simplifying these expressions often involves applying arithmetic rules and the laws of exponents, especially when dealing with fractions.
For example, the exercise involves the expression \( \frac{10 x^{4} y^{9}}{30 x^{12} y^{-3}} \). We can break this down into distinct parts to simplify:

• Simplify the coefficients: Divide \( 10 \) by \( 30 \) to get \( \frac{1}{3} \).
• Simplify each variable separately using exponent rules.

Each component, whether a number or variable, follows its respective rules. By addressing each element and simplifying step-by-step, the algebraic expression becomes much easier to manage and solve. Simplified algebraic expressions are not just a cleaner version of math problems; they are crucial in efficiently solving and understanding the broader scope of algebraic problems.