Problem 101 - Algebra 2

Question

Simplify. Rationalize all denominators. $$ \frac{2+\sqrt{10}}{2-3 \sqrt{5}} $$

Step-by-Step Solution

Answer

The final simplified equation, with the denominator rationalized, is \[(- 4\sqrt{5} + 3\sqrt{10} - 4) / 41\].

Step 1: Identify the conjugate

The conjugate of a binomial expression \(a - b\) is \(a + b\). Therefore, the conjugate of the denominator \(2 - 3\sqrt{5}\) is \(2 + 3\sqrt{5}\).

Step 2: Multiply by the conjugate

Multiply both the numerator and denominator of the given fraction by the conjugate of the denominator, which is \(2 + 3\sqrt{5}\). Thus,\[(2 + \sqrt{10}) / (2 - 3\sqrt{5}) \times (2 + 3\sqrt{5}) / (2 + 3\sqrt{5})\]

Step 3: Simplify the expression

When multiplying binomials, use FOIL (Firsts, Outers, Inners, Lasts) method. Thus, we obtain:\[(4 + 4\sqrt{5} + 3\sqrt{10}) / (4 - 45)\]

Step 4: Further simplification

The denominator simplifies to -41. Thus, the simplified fraction becomes:\[(- 4\sqrt{5} + 3\sqrt{10} - 4) / 41\].

Key Concepts

Conjugate

To rationalize a denominator, we often use the conjugate of a binomial expression. A conjugate pairs involves changing the sign between two terms in a binomial. For example, if your binomial is \(a - b\), the conjugate is \(a + b\). This helps clear out irrational numbers when they are part of a denominator.

In our original problem, the denominator is \(2 - 3\sqrt{5}\). Following the rule, its conjugate is \(2 + 3\sqrt{5}\). This process is crucial because multiplying a binomial by its conjugate gives a difference of squares, eliminating any square roots or radicals in the denominator.

Binomials

A binomial is an algebraic expression containing two terms connected by an addition or subtraction operator. For example, \(x + y\) and \(a - b\) are both binomials.

Binomials are often involved in exercises like rationalizing denominators and simplifying expressions, where understanding and using their properties can significantly help in reducing complexity.

Knowing how to manipulate binomials using operations like multiplication, especially when conjugates are involved, comes in handy. This includes using methods such as the FOIL method, which will be discussed next.

FOIL Method

The FOIL method is a technique for multiplying two binomials. FOIL stands for First, Outer, Inner, Last, which are the steps followed when multiplying. This method helps ensure that every term in one binomial multiplies with every term in the other binomial.

To elaborate, let's take the binomials \((a + b)\) and \((c + d)\). Using FOIL:

• Firsts: Multiply the first terms from each binomial: \(a \times c\)
• Outers: Multiply the outermost terms: \(a \times d\)
• Inners: Multiply the innermost terms: \(b \times c\)
• Lasts: Multiply the last terms from each binomial: \(b \times d\)

For the expression \((2 + \sqrt{10})(2 + 3\sqrt{5})\), using FOIL helps in systematically multiplying and simplifying each part of the equation as seen in the step-by-step solution provided.

Simplifying Radical Expressions

Simplifying radical expressions involves combining like terms and making expressions easier to read and interpret. This process often requires the use of conjugates, especially when there's a need to eliminate radicals from a denominator.

In our step-by-step solution: - After applying the FOIL method in the numerator, you get expressions like \(4 + 4\sqrt{5} + 3\sqrt{10}\), and in the denominator, \(4 - 45\). The terms are simplified to become more understandable.- Further simplification results in \((- 4\sqrt{5} + 3\sqrt{10} - 4) / 41\), which is your final expression, free from radicals in the denominator.

This holistic process will ensure you have a fully rationalized and simplified expression, important in mathematical accuracy and clarity.