Problem 56 - Algebra A Combined Function

Question

Rationalize each numerator. Assume that all variables represent positive real numbers. \(\frac{\sqrt{y}}{7}\)

Step-by-Step Solution

Answer

The rationalized expression is \( \frac{y}{7\sqrt{y}} \).

Step 1: Identify the Problem

We need to rationalize the numerator in the expression \( \frac{\sqrt{y}}{7} \). In this case, our numerator is \( \sqrt{y} \).

Step 2: Multiply by the Conjugate

To rationalize \( \sqrt{y} \), we multiply both the numerator and the denominator by \( \sqrt{y} \). This step utilizes the property \( a + b = a^2 \) when \( a = b \).

Step 3: Simplify the Expression

Perform the multiplication: \( \frac{\sqrt{y} \cdot \sqrt{y}}{7 \cdot \sqrt{y}} = \frac{y}{7\sqrt{y}} \). The numerator simplifies to \( y \) as \( \sqrt{y} \cdot \sqrt{y} = y \).

Step 4: Final Simplification

The expression \( \frac{y}{7\sqrt{y}} \) can be further simplified by splitting it into two separate fractions: \( \frac{y}{7} \times \frac{1}{\sqrt{y}} = \frac{y}{7\sqrt{y}} \). However, the primary task of rationalizing is complete, as there's no square root in the numerator anymore.

Key Concepts

Square Root Simplification

Understanding square root simplification is crucial when working with expressions involving roots. A square root is simply another way to ask what number, when multiplied by itself, will result in the subject number. For instance, the square root of 9 is 3 because 3 times 3 equals 9.
Below are some basics of square root simplification you should acquaint yourself with:

• Simplify each part of the radical separately. If you have a square root of a product, such as \( \sqrt{xy} \), it can be separated into \( \sqrt{x} \cdot \sqrt{y} \).
• Find perfect square factors. Simplifying \( \sqrt{18} \) involves recognizing that 18 is 9 times 2, and \( \sqrt{9} \) is 3. Thus, \( \sqrt{18} = 3\sqrt{2} \).
• A void leaving square roots in the denominator. This is because they can make the expression more complex.

Thus, when rationalizing a square root in mathematics, we aim to eliminate roots from the denominator by shifting them into the numerator.

Conjugate Multiplication

Conjugate multiplication is a handy mathematical technique, especially when dealing with the process of rationalizing. A conjugate is a binomial formed by changing the sign between two terms. For example, the conjugate of \( a + b \) is \( a - b \). In our exercise, we use a straightforward version, multiplying \( \sqrt{y} \) by itself as if it were a single term.
This method works beautifully due to the property:

• \((a + b)(a - b) = a^2 - b^2\)

This is because any square root multiplied by itself becomes just the underlying number. For instance, \( (\sqrt{y})(\sqrt{y}) = y \). It’s important to note that multiplying by the conjugate will simplify the radical expression and is a primary tool for rationalizing expressions.

Fraction Simplification

Simplifying fractions is about making them as straightforward as possible, reducing any unnecessary complexity. After following through with rationalization, you often end up with a fraction that can be further simplified.
Here are some steps to simplify fractions:

• First, check for any common factors in the numerator and denominator.
• Divide both by any common factor to simplify the expression further.
• Express fractions with a single radical in the simplest form, avoiding complex numbers in the denominator, if applicable.

In the original problem, we ended with \( \frac{y}{7\sqrt{y}} \). While it cannot be simplified further concerning simplifying fractions, rationalizing itself achieved the primary goal – removing the square root from the numerator. Remember always to simplify complex fractions by looking for common factors and reducing them.