Problem 115 - Elementary and Intermediate Algebra

Question

Simplify each expression. All variables represent positive real numbers. $$ \frac{p^{8 / 5} p^{7 / 5}}{p^{2}} $$

Step-by-Step Solution

Answer

The expression simplifies to \( p \).

Step 1: Combine the Exponents in the Numerator

When multiplying like bases, add their exponents. We start with \( p^{8/5} \times p^{7/5} \). Add the exponents: \[ \frac{8}{5} + \frac{7}{5} = \frac{15}{5} = 3. \]This simplifies the expression in the numerator to \( p^3 \).

Step 2: Simplify the Fraction

Now simplify \( \frac{p^3}{p^2} \). When dividing like bases, subtract the exponents: \[ 3 - 2 = 1. \]This gives us the simplified expression \( p^1 = p. \)

Key Concepts

Understanding Exponents

Exponents are shorthand for repeated multiplication. If you have a base, like \( p \), raised to a power, it means multiplying \( p \) by itself a number of times represented by the exponent. For example, \( p^3 \) means \( p \times p \times p \). Exponents follow specific rules, especially when they involve operations like multiplication and division. Here are some key exponent rules:

• Product Rule: When you multiply two expressions with the same base, add the exponents. For example, \( p^a \times p^b = p^{a+b} \).
• Quotient Rule: When you divide two expressions with the same base, subtract the exponents. For example, \( \frac{p^a}{p^b} = p^{a-b} \).
• Zero Exponent Rule: Any number raised to the exponent zero is one, \( p^0 = 1 \), given \( p \) is not zero.

Applying these rules helps simplify expressions efficiently, like transforming \( \frac{p^{8/5} p^{7/5}}{p^2} \) into \( p \). Understanding these concepts is crucial for managing expressions involving exponents.

Fraction Simplification Techniques

Fraction simplification involves reducing a fraction to its simplest form. In algebra, fractions often contain variables and require manipulation with exponents. To simplify a fraction:

• Identify Common Factors: Look for common factors in the numerator and the denominator.
• Cancelling Common Terms: If a term appears both in the numerator and the denominator, you can cancel these terms. For example, in \( \frac{p^3}{p^2} \), \( p^2 \) is a common part in both, simplifying to \( p^1 \).
• Using Exponent Rules: When dealing with variables, apply the exponent rules like the quotient rule. This helps in smoothly transitioning from multiple variables to a single term.

Through these steps, you maintain the integrity of the mathematical expression while simplifying it, which was key in solving the given problem.

Significance of Positive Real Numbers

Positive real numbers are all the real numbers greater than zero. In algebra and especially in the context of simplification tasks like our initial problem, working with positive real numbers allows us to apply various rules without concern for undefined or complex results.

• Consistency in Operations: With positive real numbers, operations like division and taking roots remain straightforward because these operations won't result in undefined or complex numbers.
• Positivity Implication: Knowing variables represent positive real numbers, simplifies the elimination process during fraction and exponent simplifications, as we don't deal with negative exponents turning bases into fractions.
• Simplification Approaches: Much of algebra simplifies significantly with the assumption of positive real numbers since many properties (such as commutativity and associativity) hold smoothly without additional checks for conditions.

This understanding underpins numerous algebraic processes, making calculations more approachable and often more intuitive.