Problem 370 - Elementary Algebra

Question

In the following exercises, simplify by rationalizing the denominator. $$ \frac{\sqrt{7}}{\sqrt{y}+\sqrt{3}} $$

Step-by-Step Solution

Answer

\[\frac{\text{\textbackslash sqrt\textbraceleft 7y\textbraceright}-\text{\textbackslash sqrt\textbraceleft 21\textbraceright}}{y-3}\]

Step 1: Identify the Conjugate

The first step is to identify the conjugate of the denominator. The conjugate of \(\frac{\frac{\text{\textbackslash sqrt\textbraceleft 7\textbraceright}}{\text{\textbackslash sqrt\textbraceleft y\textbraceright}+\text{\textbackslash sqrt\textbraceleft 3\textbraceright}}}\) is found by changing the sign of the second term in the denominator. Therefore, the conjugate is \(\frac{\text{\textbackslash sqrt\textbraceleft y\textbraceright}-\text{\textbackslash sqrt\textbraceleft 3\textbraceright}\).

Step 2: Multiply Numerator and Denominator by the Conjugate

Multiply both the numerator and the denominator by the conjugate identified in Step 1. This gives: \[\frac{\text{\textbackslash sqrt\textbraceleft 7\textbraceright}\times(\text{\textbackslash sqrt\textbraceleft y\textbraceright}-\text{\textbackslash sqrt\textbraceleft 3\textbraceright})}{(\text{\textbackslash sqrt\textbraceleft y\textbraceright}+\text{\textbackslash sqrt\textbraceleft 3\textbraceright})(\text{\textbackslash sqrt\textbraceleft y\textbraceright}-\text{\textbackslash sqrt\textbraceleft 3\textbraceright})}\]

Step 3: Use FOIL Method in Denominator

Use the FOIL method (First, Outer, Inner, Last) to simplify the denominator: \[ (\text{\textbackslash sqrt\textbraceleft y\textbraceright}+\text{\textbackslash sqrt\textbraceleft 3\textbraceright})(\text{\textbackslash sqrt\textbraceleft y\textbraceright}-\text{\textbackslash sqrt\textbraceleft 3\textbraceright}) = (\text{\textbackslash sqrt\textbraceleft y\textbraceright})^2 - (\text{\textbackslash sqrt\textbraceleft 3\textbraceright})^2 = y - 3 \]

Step 4: Distribute in the Numerator

Distribute the numerator across its terms: \[\text{\textbackslash sqrt\textbraceleft 7\textbraceright}\times(\text{\textbackslash sqrt\textbraceleft y\textbraceright}-\text{\textbackslash sqrt\textbraceleft 3\textbraceright}) = \text{\textbackslash sqrt\textbraceleft 7y\textbraceright} - \text{\textbackslash sqrt\textbraceleft 21\textbraceright}\]

Step 5: Write the Simplified Expression

Combine the results from Steps 3 and 4: \[\frac{\text{\textbackslash sqrt\textbraceleft 7y\textbraceright}-\text{\textbackslash sqrt\textbraceleft 21\textbraceright}}{y-3}\]

Key Concepts

Conjugate

In algebra, a conjugate is formed by changing the sign between two terms. This means if you have a term like \(\text{a} + \text{b}\), its conjugate would be \(\text{a} - \text{b}\). Conjugates are very useful when we need to rationalize denominators, especially when dealing with radicals. To eliminate a radical in the denominator, we multiply the expression by the conjugate of the denominator. This works because multiplying a binomial by its conjugate results in the difference of squares, a method that effectively removes the radical.

FOIL Method

The FOIL method is an acronym standing for First, Outer, Inner, Last. It's a way to multiply two binomials. For example, when multiplying \((a+b)\cdot(c+d)\) using FOIL, you perform the following:

• First: Multiply the first terms \((a \cdot c)\)
• Outer: Multiply the outer terms \((a \cdot d)\)
• Inner: Multiply the inner terms \((b \cdot c)\)
• Last: Multiply the last terms \((b \cdot d)\)

Finally, you combine these products. In the context of rationalizing the denominator, using FOIL method transforms the product of a binomial and its conjugate into easier, non-radical expressions.

Simplifying Radicals

Simplifying radicals involves expressing a radical in its simplest form. This process might include:

• Finding the prime factors of the number inside the radical.
• Pairing the factors to simplify the radical.

For example, to simplify \(\sqrt{18}+\sqrt{32}\), find the prime factorization:
\(18 = 2 \cdot 3 \cdot 3 \), so \(\sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2}\).
\(32 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 4 \cdot 8\)= so \(\sqrt{32} = \sqrt{16 \cdot 2} = 4\sqrt{2}\).
Simplifying radicals is key when you need to simplify algebraic expressions that involve them.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They can be as simple as \(x+5\) or more complex like \(\frac{\sqrt{7}}{\sqrt{y}+\sqrt{3}}\). The key to working with algebraic expressions is to understand the rules and properties of operations, like addition, subtraction, multiplication, and division.
In rationalizing the denominator, we used several algebraic principles:

• The principle of multiplying by the conjugate.
• Using the FOIL method to expand products.
• Simplifying resulting radicals.
• Combining terms appropriately.

Understanding these rules helps in manipulating and simplifying complex expressions effectively.