Problem 65 - Algebra 1

Question

$$\text { Simplify } \frac{3}{5-\sqrt{5}}$$ \(\begin{array}{llll}\text { (A) } \frac{15+3 \sqrt{5}}{20} & \text { (B) } \frac{15+\sqrt{5}}{20} & \text { (C) } \frac{15+\sqrt{15}}{20}\end{array}\) (D) \(\frac{15-3 \sqrt{5}}{20}\)

Step-by-Step Solution

Answer

(A) \(\frac{15+3 \sqrt{5}}{20}\)

Step 1: Multiply by the Conjugate

To deal with the radical in the denominator, the expression is multiplied by the conjugate of the denominator over itself, i.e. \((5+\sqrt{5})/(5+\sqrt{5})\).

Step 2: Simplify the Numerator

The numerator becomes \(3*(5+\sqrt{5}) = 15+3\sqrt{5}\).

Step 3: Simplify the Denominator

The denominator becomes \((5-\sqrt{5})*(5+\sqrt{5}) = 5^2 - (\sqrt{5})^2 = 25 - 5 = 20\).

Step 4: Write the Simplified Fraction

The simplified fraction is \((15+3\sqrt{5})/20\).

Key Concepts

Simplifying Expressions

Simplifying expressions means reducing them into their most concise and understandable form. When you're given a fraction with a radical in the denominator, it's often necessary to "simplify" the expression to make it mathematically cleaner. The ultimate goal is to have a rational number in the denominator.

For example, take \( \frac{3}{5-\sqrt{5}} \). Initially, this fraction isn't considered fully simplified because the radical, \( \sqrt{5} \), is in the denominator. This is where the process of multiplying by the conjugate comes into play. Simplifying expressions often requires manipulation, such as multiplying by conjugates, to remove radicals from the denominator.

Remember, the main focus is to ensure the expression is in its simplest form, making it easier to work with in more extensive calculations.

Conjugate Pairs

Conjugate pairs are two binomial expressions that have opposite signs between their terms. They are incredibly useful when it comes to rationalizing the denominator. For example, if you have \(5-\sqrt{5}\), its conjugate would be \(5+\sqrt{5}\).

Multiplying conjugate pairs results in the difference of squares, effectively eliminating radicals. When rationalizing a denominator like \(5-\sqrt{5}\), you multiply both the numerator and denominator by \(5+\sqrt{5}\). This process not only helps to simplify the expression but also gets rid of the radical in the denominator.

Using conjugates keeps the value of the fraction unchanged since you're multiplying by a form of 1, namely \(\frac{5+\sqrt{5}}{5+\sqrt{5}}\). This is an essential technique in algebra for simplifying complex fractions involving roots.

Radical Expressions

Radical expressions include roots, like square roots, and often make expressions more complex if they appear in the denominator. Learning to handle them is crucial in algebra. They need special care, especially when found in the denominator, because a radical in the denominator isn't considered simplified.

To manage radicals, you can use techniques like squaring or using conjugates. In our example, \(\sqrt{5}\) in the denominator was managed using a conjugate to simplify the fraction \(\frac{3}{5-\sqrt{5}}\). The goal is to eliminate the radical from the denominator without changing the value of the expression itself.

Understanding how to work with radical expressions will greatly enhance your algebra skills and is vital for tackling a wide variety of problems.