Question

Simplify each expression. \(\frac{1}{\sqrt{5}}\)

Short answer

\(\frac{\sqrt{5}}{5}\)

Step-by-step solution

Identify the Need for Rationalization

When simplifying expressions like \(\frac{1}{\sqrt{5}}\), we look to rationalize the denominator so that it does not include a square root. This is necessary because having square roots in denominators is not considered a simplified form.

Rationalize the Denominator

To eliminate the square root in the denominator, multiply both the numerator and the denominator by \(\sqrt{5}\). This will result in the expression: \(\frac{1 \cdot \sqrt{5}}{\sqrt{5} \cdot \sqrt{5}}\).

Simplify the Expression

Now perform the multiplication: \(\sqrt{5} \cdot \sqrt{5} = 5\). Hence, the expression becomes \(\frac{\sqrt{5}}{5}\). This is the simplified version of the original expression.

Key concepts

Simplifying Expressions

When we talk about simplifying expressions, the goal is to transform an expression into its simplest form. This means there should be no square roots or complex numbers in the denominator. Usually, a simpler form is easier to interpret or compute with. For expressions like \(\frac{1}{\sqrt{5}}\), having a square root in the denominator is not considered simplified. This is because dividing by a non-integer, especially when it's under a root, can create complications in further computations or when communicating results. A crucial aspect of simplifying is recognizing when an expression meets or violates these criteria. This often leads to techniques like rationalizing the denominator, which we'll cover next.

Square Roots

Square roots can often feel a little tricky, but they're essentially asking "What number, when multiplied by itself, gives us this number?" For example, the square root of 25, written as \(\sqrt{25}\), is 5, since 5 times 5 equals 25. Understanding square roots goes a long way in simplifying mathematical expressions. When the square root exists in the denominator (the bottom part of a fraction), it's usually an indicator that the expression is not in its simplest form. By multiplying the denominator by itself, a technique known as rationalization, we can eliminate the root. This not only simplifies the expression but makes future computations smoother and more understandable.

Mathematical Operations

Involved in simplifying expressions is the application of basic mathematical operations: addition, subtraction, multiplication, and division. Each operation follows specific rules and can impact the way we simplify expressions. For example, in rationalizing \(\frac{1}{\sqrt{5}}\), we use multiplication. By multiplying both the numerator and the denominator by \(\sqrt{5}\), we effectively eliminate the square root from the denominator. Operations must also respect the properties of numbers, such as the commutative and associative properties, particularly when dealing with square roots. This ensures that your computations remain accurate and conform to mathematical standards. Recognizing which operations to perform and when is key to mastering the art of expression simplification.