Problem 24 - Elementary and Intermediate Algebra

Question

$$ \frac{-35 x z^{6}}{-7 x^{8} z^{2}} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{5z^4}{x^7}\).

Step 1: Identify the Expression

The given expression is \(\frac{-35xz^6}{-7x^8z^2}\). We notice that the numerator is \(-35xz^6\) and the denominator is \(-7x^8z^2\).

Step 2: Simplify the Coefficients

Divide the coefficients \(-35\) and \(-7\) by \(-7\). The result is \(\frac{-35}{-7} = 5\).

Step 3: Simplify the Variables

Apply the property of exponents \(\frac{x^m}{x^n} = x^{m-n}\) to the \(x\) terms: \(\frac{x}{x^8} = x^{1-8} = x^{-7}\).

Step 4: Simplify the \(z\) Terms

Use the same property for the \(z\) terms: \(\frac{z^6}{z^2} = z^{6-2} = z^4\).

Step 5: Combine the Results

Combine the simplified components: \(5x^{-7}z^4\).

Step 6: Express with Positive Exponents

Convert \(x^{-7}\) with positive exponents: \(x^{-7} = \frac{1}{x^7}\). Thus, we rewrite the expression as \(\frac{5z^4}{x^7}\).

Key Concepts

Properties of Exponents

Exponent properties are essential to simplifying expressions that involve powers of a number or variable. They provide a set of rules that determine how exponents should be handled in mathematical operations. The rules are especially useful for simplifying complex expressions. Here's a closer look:

• **Product of Powers**: When multiplying two expressions with the same base, add the exponents: \[a^m \times a^n = a^{m+n}\]
• **Quotient of Powers**: When dividing two expressions with the same base, subtract the exponents: \[\frac{a^m}{a^n} = a^{m-n}\]
• **Power of a Power**: When raising an expression already containing an exponent to another power, multiply the exponents: \[(a^m)^n = a^{m\times n}\]

These properties make it simpler to evaluate and reduce expressions involving exponents. In the given exercise, we predominantly use the "Quotient of Powers" property to simplify the variable sections of the expression, transforming complex parts into manageable terms. This is the reason why we could reduce \(\frac{x}{x^8}\) to \(x^{-7}\). Simplifying these expressions initially might seem tough but practicing these properties helps embed these so you can handle any exercise!

Simplifying Algebraic Fractions

Algebraic fractions are similar to numerical fractions, but they contain variables. They demand a good understanding of both basic algebra and fraction rules. The primary goal when dealing with algebraic fractions is to simplify the expression to its lowest form for easier interpretation and further manipulation.When simplifying algebraic fractions, follow these steps:

• **Factor Coefficients**: Start by factoring both the numerator and denominator. In our exercise, we divided the coefficients \(-35\) and \(-7\) yielding \(5\).
• **Apply Exponents Rules**: Use properties of exponents to simplify variables. This involves canceling like terms and reducing them using "Quotient of Powers". Our example reduced the x terms to \(x^{-7}\) and the z terms to \(z^4\).
• **Rewrite the Expression**: Finally, rewrite the expression combining all simplified parts, leading to \(\frac{5z^4}{x^7}\).

By breaking down the fraction into manageable parts through these steps, you gain control over its complexity. This approach not only simplifies calculations but also provides clearer insights into the composition of the fraction.

Negative Exponents

Negative exponents can seem confusing at first, but they follow a specific pattern that makes them easier to work with when simplifying expressions. The role of a negative exponent is to represent the reciprocal of the base raised to the equivalent positive exponent. Here’s a breakdown of how to manage them:

• **Reciprocal Rule**: A negative exponent signifies a reciprocal, hence \[a^{-m} = \frac{1}{a^m}\]. This is particularly important when dealing with division in expressions, as seen in the simplification from \(x^{-7}\) to \(\frac{1}{x^7}\).
• **Simplifying Expressions**: When simplifying, aim to rewrite expressions with negative exponents as fractions with positive exponents. This adjustment converts complex expressions into simpler forms that are easier to work with.
• **Recognition**: Recognize that negative exponents don’t make the value itself negative but change how the base is manipulated by inversing it.

Understanding negative exponents enhances your ability to simplify and manipulate complex fractions. This is and will be a useful skill for algebra, calculus, and beyond. Keep in mind the idea of reciprocity, and you will always know how to tackle tricky expressions.