Problem 53 - Algebra A Combined Function

Question

Rationalize each numerator. Assume that all variables represent positive real numbers. \(\sqrt{\frac{2}{5}}\)

Step-by-Step Solution

Answer

The rationalized form is \( \frac{\sqrt{10}}{5} \).

Step 1: Problem Recollection

We are given \( \sqrt{\frac{2}{5}} \) and asked to rationalize the numerator.

Step 2: Rationalization Purpose

Rationalizing the numerator involves rewriting the expression to eliminate any square roots from the numerator.

Step 3: Multiply by Conjugate

Multiply both numerator and denominator by the square root of the denominator, \(\sqrt{5}\). This is because multiplying by the conjugate of the denominator will rationalize the expression: \( \sqrt{\frac{2}{5}}\times\frac{\sqrt{5}}{\sqrt{5}} = \frac{\sqrt{2} \times \sqrt{5}}{\sqrt{5} \times \sqrt{5}} \).

Step 4: Simplify the Expression

Simplify the expression obtained: \( \frac{\sqrt{10}}{5} \). This is because \( \sqrt{5}\times\sqrt{5} = 5\).

Step 5: Verify the Result

The expression \( \frac{\sqrt{10}}{5} \) has no square roots in the numerator, confirming it is rationalized.

Key Concepts

Understanding Square Roots

Square roots are fundamental in mathematics. A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, because 3 times 3 equals 9. When you see the symbol \( \sqrt{\cdot} \), it denotes the principal square root, meaning the non-negative root. Square roots can be challenging because they often result in irrational numbers—those numbers that cannot be expressed as a simple fraction. When rationalizing expressions, especially in algebra, it's essential to deal with these square roots effectively.

• Square roots simplify to integer values for perfect squares (e.g., 4, 9, 16).
• Non-perfect square roots, like \( \sqrt{2} \), are irrational numbers.
• They often appear in roots of fractions or radical expressions.

Grasping the concept of square roots is vital for various operations, such as rationalizing numerators and denominators in fractions.

The Role of the Numerator in Fraction Rationalization

In mathematical expressions, the numerator is the top part of a fraction. For instance, in the fraction \( \frac{2}{5} \), the numerator is 2. When dealing with square roots, the presence of a radical in the numerator can make it more difficult to manipulate and understand the expression. That's why there's often a need to rationalize it.

• Rationalizing involves removing the square root from the numerator.
• This process can simplify calculations by making expressions easier to handle.
• The goal is to convert the radical expression into a form without a square root in the numerator.

Rationalized expressions are often preferred because they can be simpler and convey clearer mathematical relationships.

Effective Rationalization Techniques

Rationalization techniques are a set of methods used to eliminate irrational components from the numerator or denominator of a fraction. One common method is multiplying by the conjugate. Here's how it works:

• Identify the square root in the numerator that needs to be removed.
• Multiply both the numerator and the denominator by a suitable radical that will simplify the expression, often the denominator itself.

In the example \( \sqrt{\frac{2}{5}} \), multiplying by \( \frac{\sqrt{5}}{\sqrt{5}} \) helps in removing the square root from the numerator:

• This transforms the expression to \( \frac{\sqrt{10}}{5} \).
• The square root of 5 in the denominator becomes a regular number as \( \sqrt{5} \times \sqrt{5} = 5 \).

By effective application of these techniques, expressions are simplified, making further mathematical operations more accessible and straightforward.