Problem 34 - Algebra 2

Question

Simplify. Rationalize all denominators. Assume that all the variables are positive. $$ (2 \sqrt{5}+3 \sqrt{2})(5 \sqrt{5}-7 \sqrt{2}) $$

Step-by-Step Solution

Answer

The simplified form of the expression \( (2 \sqrt{5}+3 \sqrt{2})(5 \sqrt{5}-7 \sqrt{2}) \) is \( 8 + \sqrt{10} \)

Step 1: Expand the Product

First, expand the product using the distributive property, also known as the FOIL method.\n\n\( (2 \sqrt{5} \cdot 5 \sqrt{5}) + (2 \sqrt{5} \cdot -7 \sqrt{2}) + (3 \sqrt{2} \cdot 5 \sqrt{5}) + (3 \sqrt{2} \cdot -7 \sqrt{2}) \)

Step 2: Simplify the Results of Multiplication

Next, simplify the four terms obtained in step 1. Remember that the product of \( \sqrt{a} \) and \( \sqrt{a} \) is a.\n\nThe results are: \( (2 \cdot 5 \cdot 5) + (2 \cdot -7 \cdot \sqrt{10}) + (3 \cdot 5 \cdot \sqrt{10}) + (3 \cdot -7 \cdot 2) \)

Step 3: Compute the Results

Finally, perform the arithmetic operations to derive the simplified expression: \( 50 - 14 \sqrt{10} + 15 \sqrt{10} - 42 = 8 + \sqrt{10} \)

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that helps simplify expressions, especially when dealing with products of sums. When you come across an expression like \((a + b)(c + d)\),you can apply the distributive property to expand it into\(ac + ad + bc + bd\).This is often referred to as the FOIL method when dealing with two binomials, although the underlying principle is the same.
In the given exercise,\((2 \sqrt{5} + 3 \sqrt{2})(5 \sqrt{5} - 7 \sqrt{2})\),we apply the distributive property to each term in the first set of parentheses, multiplying it by each term in the second set. This breaks down the problem into smaller, more manageable multiplication tasks, which we can then solve step-by-step.
By systematically applying the distributive property,

• First, multiply the first terms: \(2 \sqrt{5} \times 5 \sqrt{5}\).
• Next, the outer terms: \(2 \sqrt{5} \times -7 \sqrt{2}\).
• Then the inner terms: \(3 \sqrt{2} \times 5 \sqrt{5}\).
• Finally, the last terms: \(3 \sqrt{2} \times -7 \sqrt{2}\).

You end up with four expressions, which can then be simplified through additional multiplication and combining any like terms.

Rationalizing Denominators

Rationalizing the denominator is a technique used to eliminate radicals from the denominator of a fraction. It makes expressions easier to handle and more aesthetically pleasing. Typically, we use this process whenever a denominator has a radical involved.
Even though the original exercise does not specifically mention fractions, rationalizing is a necessary skill when dealing with more complex algebraic expressions that might have denominator radicals.
To perform rationalization:

• Multiply the numerator and denominator by a convenient form of 1 that will eliminate the radical in the denominator, such as the radical itself or its conjugate.
• Simplify the multiplied terms, aiming to have a rational number in the denominator.

For example, if you have \(\frac{1}{\sqrt{2}}\), rationalize by multiplying top and bottom by \(\sqrt{2}\) to get\(\frac{\sqrt{2}}{2}\).
Though not needed directly in our exercise, understanding this concept is critical for handling radicals thoroughly.

FOIL Method

The FOIL method is a specific application of the distributive property tailored for multiplying two binomials. FOIL is an acronym representing First, Outside, Inside, and Last, and it dictates the order in which you multiply the terms.
With our given problem,\((2 \sqrt{5} + 3 \sqrt{2})(5 \sqrt{5} - 7 \sqrt{2})\),we use FOIL to ensure all combinations of terms are covered:

• **First:** Multiply the first terms in each binomial, resulting in \(2 \sqrt{5} \times 5 \sqrt{5}\).
• **Outside:** Multiply the outer terms, resulting in \(2 \sqrt{5} \times -7 \sqrt{2}\).
• **Inside:** Multiply the inner terms, resulting in \(3 \sqrt{2} \times 5 \sqrt{5}\).
• **Last:** Multiply the last terms, resulting in \(3 \sqrt{2} \times -7 \sqrt{2}\).

Each of these products should be calculated individually, and the final step is to combine all these results into a single simplified expression. FOIL is especially useful because it ensures that no term is left unmultiplied, which could lead to inaccuracies in simplification if overlooked.