Problem 47 - Algebra 1

Question

Simplify the expression. $$\frac{12}{7-\sqrt{3}}$$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{42 + 6\sqrt{3}}{23} \).

Step 1: Identify the Conjugate

In this step, identify the conjugate of the denominator of the given expression. The conjugate of \(7-\sqrt{3}\) is \(7 + \sqrt{3}\).

Step 2: Multiply by the Conjugate

In this step, multiply the numerator and the denominator by the conjugate. So, \(\frac{12}{7 - \sqrt{3}}\) multiply by \(\frac{7 + \sqrt{3}}{7 + \sqrt{3}}\) to get \(\frac{12 (7 + \sqrt{3})}{(7 - \sqrt{3})(7 + \sqrt{3})}\).

Step 3: Simplify the Expression

In this step, simplify the expression by applying the difference of squares formula in the denominator which states that \(a^2 - b^2 = (a - b)(a + b)\), and distribute the 12 in the numerator. This gives \(\frac{84 + 12\sqrt{3}}{49 - 3} = \frac{84 + 12\sqrt{3}}{46}\).

Step 4: Further Simplify

The numerator and the denominator have a common factor 2, so we can divide both sides by 2. This gives the simplified expression \(\frac{42 + 6\sqrt{3}}{23}\).

Key Concepts

Conjugates

When working with expressions that include roots in the denominator, one method to simplify the expression is to "rationalize the denominator." This often involves using the concept of conjugates. A conjugate is formed by changing the sign between two terms. For example, the conjugate of the expression

• \( 7 - \sqrt{3} \) is \( 7 + \sqrt{3} \).

Multiplying the original expression's numerator and denominator by this conjugate helps eliminate the square root from the denominator. This is because when a binomial expression, such as \((a - b)\), is multiplied by its conjugate, \((a + b)\), the result is a difference of squares.

Simplifying Expressions

Simplifying mathematical expressions is a crucial step in solving equations and making calculations easier. In our example, we simplify \( \frac{12(7+\sqrt{3})}{(7-\sqrt{3})(7+\sqrt{3})} \) by distributing the numerator and applying algebraic identities.

• First, distribute the 12 across the terms inside the parenthesis: \( 12 \times 7 + 12 \times \sqrt{3} \).
• This results in \( 84 + 12\sqrt{3} \).

It's essential to simplify expressions because it makes them easier to understand and work with, especially when solving equations. In many cases, after distribution and simplification, you might notice common factors that allow further reduction of the expression.

Difference of Squares

The difference of squares is a specific algebraic pattern used to simplify expressions like the ones we encounter when rationalizing denominators. The identity is given by:

• \( a^2 - b^2 = (a - b)(a + b) \)

Applying this rule to

• \( (7 - \sqrt{3})(7 + \sqrt{3}) \)

results in a simpler expression:

• \( 7^2 - (\sqrt{3})^2 = 49 - 3 = 46 \).

Understanding this pattern is essential when dealing with polynomial expressions and helps in simplifying rational expressions by eliminating radicals, streamlining calculations, and leading the way to a more straightforward solution.