Problem 51 - Algebra 1: Concepts and Skills

Question

Simplify the radical expression. $$ \frac{\sqrt{5}}{3-\sqrt{5}} $$

Step-by-Step Solution

Answer

The simplified form of the expression is \(\frac{3\sqrt{5} + 5} {4}\)

Step 1: Identify and rewrite the expression

After understanding the problem, the radical expression can be written as \(\frac{\sqrt{5}}{3-\sqrt{5}}\)

Step 2: Rationalize the denominator

Rationalizing the denominator involves multiplying both numerator and denominator by the conjugate of the denominator. Here, the conjugate of \(3-\sqrt{5}\) is \(3+\sqrt{5}\). Thus the expression becomes \(\frac{\sqrt{5}(3+\sqrt{5})} {(3-\sqrt{5})(3+\sqrt{5})}\)

Step 3: Expand the terms in the numerator and denominator

For the numerator: \(\sqrt{5}*3 + \sqrt{5}*\sqrt{5}\) which can be simplified to \(3\sqrt{5} + 5\). For the denominator, it's a perfect square of a binomial which solves to \(3^2 - (\sqrt{5})^2\), which simplifies to \(9-5\)

Step 4: Simplify the expression

The expression now becomes \(\frac{3\sqrt{5} + 5} {4}\)

Key Concepts

Radical Expressions

A radical expression includes a square root (or other roots), and in our problem, involves the square root of 5. Understanding radical expressions is key to solving problems where we need to simplify or manipulate square roots.

A square root is a factor of a number that, when multiplied by itself, gives the original number. For instance, in \(\sqrt{5}\), the expression represents the number which, when squared, equals 5.

Working with radicals involves several operations:

• Adding or subtracting radicals requires them to have the same radicand and index, which is similar to combining like terms in algebra.
• Multiplying and dividing radicals is often simpler, as different radicands can be multiplied or divided, and then simplified if possible.

Radical expressions come into play in various mathematical operations, especially when rationalizing denominators, as they often contain roots. Simplifying these expressions is crucial for clearer mathematical communication and understanding.

Conjugates

Conjugates are an essential concept in rationalizing the denominator, especially with expressions involving square roots. In mathematics, the conjugate of a binomial is derived by changing the sign between two terms. For example, the conjugate of \(a - b\) is \(a + b\).

When dealing with a denominator like \(3-\sqrt{5}\), we use its conjugate, \(3+\sqrt{5}\), to multiply the numerator and denominator of the fraction. The multiplication with its conjugate helps, because:

• The product of a binomial and its conjugate yields a difference of squares, eliminating the radical in the denominator.
• It results in a rational number, making the fraction easier to work with.

This method makes solutions more straightforward and canonical, ensuring the expression is in its simplest form. This process also helps us to better visualize and solve complex expressions involving radicals.

Simplifying Fractions

Simplifying fractions is a fundamental process in mathematics, allowing us to express a fraction in its simplest form. Doing this involves reducing a fraction to its lowest terms or ensuring there are no remaining radicals or complex numbers in the denominator.

In our current problem, after multiplying by the conjugate, we reach a simplified form: \(\frac{3\sqrt{5} + 5}{4}\). Here, every part of the expression has been simplified:

• The numerator involves straightforward arithmetic with radicals, resulting in terms like \(3\sqrt{5}\) and 5.
• The denominator becomes a straightforward whole number, as the square root is effectively removed through conjugate multiplication.

Simplifying fractions, particularly with radical expressions, ensures accuracy and efficiency in solving mathematical problems. It aids in clearer communication of results and eases further computations or integrations of these values in different contexts.