Problem 121 - Elementary and Intermediate Algebra

Question

Simplify each expression. All variables represent positive real numbers. $$ n^{1 / 5}\left(n^{2 / 5}-n^{-1 / 5}\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(n^{3/5} - 1\).

Step 1: Distribute the Expression

Distribute the term \(n^{1/5}\) across the expression inside the parentheses \(n^{2/5} - n^{-1/5}\). This means that you multiply \(n^{1/5}\) by each term separately:\[n^{1/5} \cdot n^{2/5} - n^{1/5} \cdot n^{-1/5}\]

Step 2: Applying the Exponent Rule

Recall the exponent multiplication rule \(a^m \cdot a^n = a^{m+n}\). Apply this rule to each part of the expression from Step 1:1. \(n^{1/5} \cdot n^{2/5} = n^{(1/5) + (2/5)} = n^{3/5}\)2. \(n^{1/5} \cdot n^{-1/5} = n^{(1/5) + (-1/5)} = n^{0} = 1\)

Step 3: Write the Simplified Expression

Combine the results from Step 2 into a single expression:\[n^{3/5} - 1\]This is the simplified form of the original expression.

Key Concepts

Exponent Rules

Exponent rules are vital when simplifying algebraic expressions. They help us manage the powers of numbers in a systematic way. Understanding these rules offers a straightforward pathway to simplify and solve equations that involve powers.

One of the primary rules is the multiplication of exponents. When you multiply two powers with the same base, you add their exponents. This rule can be written as: \(a^m \cdot a^n = a^{m+n}\).

• For example, if you have \(n^{1/5} \cdot n^{2/5}\), you add the exponents \((1/5)+ (2/5)\) which equals \(3/5\). So, the result is \(n^{3/5}\).
• Another principal rule is that any number raised to the power of zero equals 1: \(a^0 = 1\). In our example, \(n^{1/5} \cdot n^{-1/5}\) becomes \(n^0\), simplifying to 1.

Understanding and mastering these rules make the simplification of expressions a much more manageable task.

Distributive Property

The distributive property is a fundamental algebra concept that is used to simplify expressions and make calculations easier. It applies when you have a term being multiplied by a group of terms inside parentheses.

Mathematically, this property is expressed as: \(a(b + c) = ab + ac\). In this rule, we distribute \(a\) across each term inside the parentheses.

• In our expression \(n^{1/5}(n^{2/5} - n^{-1/5})\), the distributive property means we multiply \(n^{1/5}\) with each term inside the parentheses individually: \(n^{1/5} \cdot n^{2/5}\) and \(n^{1/5} \cdot n^{-1/5}\). This helps in breaking down complex expressions into simpler parts.
• By distributing the terms, it allows us to use other mathematical principles, like exponent rules, to simplify the expression further.

The distributive property is quite powerful and often used in simplifying and solving equations in algebra.

Positive Real Numbers

Throughout the original exercise, it was explicitly stated that all variables represent positive real numbers. This is important because it ensures certain mathematical properties are consistent, such as those related to exponents and the simplification processes.

Positive real numbers are numbers greater than zero and do not include any imaginary or complex numbers.

• Operations involving positive real numbers, such as exponentiation, always yield predictable results. For instance, raising a positive real number to any power will not change its positivity.
• When working with exponents, it's crucial to know that the base is positive, which maintains the validity of simplification rules like \(a^0 = 1\).
• This assumption also simplifies expressions in algebra by preventing concerns about negative bases leading to undefined situations.

Understanding these properties ensures the calculations stay within the realm of real numbers and maintain the results' accuracy and validity.