Problem 21 - Algebra 2 and Trigonometry

Question

In \(3-29\) write each quotient in simplest form. Variables in the radicand with an even index are non-negative. Variables occurring in the denominator of a fraction are non-zero. $$ \frac{\sqrt{20}-\sqrt{5}}{\sqrt{5}} $$

Step-by-Step Solution

Answer

The simplified expression is 1.

Step 1: Simplify the Numerator

First, simplify the expression by working on the numerator. Look at \( \sqrt{20} \) and \( \sqrt{5} \). Rewrite \( \sqrt{20} \) as \( \sqrt{4 \times 5} = \sqrt{4} \cdot \sqrt{5} = 2\sqrt{5} \). This changes the numerator from \( \sqrt{20} - \sqrt{5} \) to \( 2\sqrt{5} - \sqrt{5} \).

Step 2: Factor the Numerator

Factor the expression in the numerator by taking \( \sqrt{5} \) common: \( 2\sqrt{5} - \sqrt{5} = \sqrt{5}(2 - 1) = \sqrt{5} \cdot 1 = \sqrt{5} \). The numerator simplifies to \( \sqrt{5} \).

Step 3: Combine and Simplify the Expression

Combine the simplified numerator with the denominator. This gives us \( \frac{\sqrt{5}}{\sqrt{5}} \).

Step 4: Simplify the Fraction

Since the numerator and denominator are identical, \( \frac{\sqrt{5}}{\sqrt{5}} = 1 \). So the expression simplifies to 1.

Key Concepts

Numerical Expressions

Numerical expressions are mathematical phrases that involve only numbers and operations such as addition, subtraction, multiplication, and division. Understanding how to simplify these expressions is crucial for tackling more complex mathematical problems. In the provided exercise, the expression is given as a fraction with numerators and denominators involving square roots:

• The numerator is initially given as \( \sqrt{20} - \sqrt{5} \).
• To simplify \( \sqrt{20} \), we recognize it as \( \sqrt{4 \times 5} \), which further breaks down to \( \sqrt{4} \times \sqrt{5} = 2\sqrt{5} \).
• This allows us to express the numerator clearly as \( 2\sqrt{5} - \sqrt{5} \).

Subsequently, by simplifying the numerical expression, it becomes easier to approach the overall problem, reducing errors in further calculations.

Factoring

Factoring in mathematics is the process of breaking down an expression into a product of simpler expressions. In the context of this problem, factoring is used to simplify the numerator. After simplifying \( \sqrt{20} \), we have the expression \( 2\sqrt{5} - \sqrt{5} \). Here, factoring plays a crucial role:

• We notice that \( \sqrt{5} \) is common in both terms of the expression \( 2\sqrt{5} - \sqrt{5} \).
• By factoring \( \sqrt{5} \) out, we rewrite the expression as \( \sqrt{5}(2 - 1) \).
• This simplifies further to \( \sqrt{5} \cdot 1 = \sqrt{5} \).

Factoring is a valuable technique because it allows us to recognize and simplify common components within a mathematical expression, thereby making computation much more straightforward.

Rational Expressions

Rational expressions involve fractions where the numerator and/or the denominator contain polynomials. Although our given exercise primarily involves radicals, it represents a basic form of rational expressions:

• The expression \( \frac{\sqrt{5}}{\sqrt{5}} \) is equivalent to \( \frac{a}{a} \) form, where \( a = \sqrt{5} \).
• In mathematics, when the numerator and the denominator of a fraction are identical and non-zero, the expression simplifies to 1.
• This demonstrates a fundamental property of rational expressions, where equivalent expressions simplify to a constant.

Understanding these properties of rational expressions helps make simplifying radicals not just manageable, but intuitive, as in this scenario, where the complex-looking expression simplifies elegantly to 1.