Problem 3 - Algebra A Combined Function

Question

Rationalize each denominator. Assume that all variables represent positive real numbers. \(\sqrt{\frac{1}{5}}\)

Step-by-Step Solution

Answer

The rationalized form is \( \frac{\sqrt{5}}{5} \).

Step 1: Understand the Problem

We need to rationalize the denominator of the expression \( \sqrt{\frac{1}{5}} \). This means we want to rewrite the expression so that there is no square root in the denominator.

Step 2: Rewrite with Common Square Root

The expression \( \sqrt{\frac{1}{5}} \) can be rewritten as \( \frac{\sqrt{1}}{\sqrt{5}} \). This separates the numerator and the denominator under the square root.

Step 3: Identify the Rationalizing Factor

To eliminate the square root in the denominator, we multiply both the numerator and the denominator by \( \sqrt{5} \), the square root of the denominator: \( \frac{\sqrt{1}}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} \).

Step 4: Perform Multiplication

Multiply the numerators and the denominators: \( \frac{\sqrt{1} \times \sqrt{5}}{\sqrt{5} \times \sqrt{5}} = \frac{\sqrt{5}}{\sqrt{25}} \).

Step 5: Simplify the Denominator

The denominator becomes \( \sqrt{25} \), which simplifies to \( 5 \). So the expression is now \( \frac{\sqrt{5}}{5} \).

Step 6: Final Expression

The rationalized form of \( \sqrt{\frac{1}{5}} \) is \( \frac{\sqrt{5}}{5} \).

Key Concepts

Square Roots

Square roots are mathematical operations that identify a number which, when multiplied by itself, yields the original number. For example, the square root of 25 is 5 because 5 times 5 equals 25. Symbolically, the square root is represented by the radical sign "\( \sqrt{ } \)".

• The content under the radical is known as the "radicand".
• In mathematics, operations often involve "simplifying" square roots for ease of handling equations.

When it comes to rationalizing denominators, understanding square roots is essential. The main goal is to remove any square roots from the denominator of a fraction to make the expression look cleaner and easier to interpret.

Simplifying Expressions

Simplifying expressions is all about making an expression as simple as possible. This process helps in making calculations easier. When you have a fraction with a square root in the denominator, simplifying it by rationalizing the denominator is a common practice.

• Separate the numerator and denominator under the radical when necessary.
• Multiply by a form of 1 that will eliminate the square root from the denominator.

For example, in the problem \(\sqrt{\frac{1}{5}}\), the expression \( \frac{\sqrt{1}}{\sqrt{5}} \) was simplified by multiplying by \( \sqrt{5}/\sqrt{5} \) to help simplify the expression to \( \frac{\sqrt{5}}{5} \). This is now in a simple and easily manageable form.

Multiplying Radicals

Multiplying radicals involves combining two or more terms under the same root sign. This often means dealing with expressions like \( \sqrt{a} \times \sqrt{b} = \sqrt{a \times b} \). It is crucial when rationalizing denominators as it helps remove the square root from the denominator.

• The product of a radical term with itself results in a non-radical term: \( \sqrt{x} \times \sqrt{x} = x \).
• This property is useful in simplifying expressions with radicals in the denominator.

In the given problem, \( \frac{\sqrt{1}}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} \) illustrates how multiplying with the same radical term eliminates the square root in the denominator, simplifying it to a more basic number like 5.