Problem 71 - Algebra 2

Question

Multiply. \((3 \sqrt{5}+2 \sqrt{10})(2 \sqrt{5}+\sqrt{10})\)

Step-by-Step Solution

Answer

The simplified and condensed result of multiplying \((3 \sqrt{5}+2 \sqrt{10})(2 \sqrt{5}+\sqrt{10})\) is \(50 + 30\sqrt{2} + 20 \sqrt{10}\).

Step 1: Expand the expression.

Begin by expanding the given expression as you would with any binomial. For \((a+b)(c+d)\), multiply \((a \cdot c)\), \((a \cdot d)\), \((b \cdot c)\), and \((b \cdot d)\) separately and add the results to give \((a \cdot c) + (a \cdot d) + (b \cdot c) + (b \cdot d)\). In our case, this gives \((3 \sqrt{5} \cdot 2 \sqrt{5}) + (3 \sqrt{5} \cdot \sqrt{10}) + (2 \sqrt{10} \cdot 2 \sqrt{5}) + (2 \sqrt{10} \cdot \sqrt{10})\).

Step 2: Simplify the multiplication of radicals

Simplify each of the multiplication separately. Remember \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\) and \(\sqrt{a} \cdot \sqrt{a} = a\). This gives: \(30 + 6 \sqrt{50} + 20 \sqrt{10} + 20\).

Step 3: Simplify the radicals.

Simplify the radical form by finding perfect square factors of the number under the radical. \( \sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2}\), giving us \(30 + 6 \cdot 5\sqrt{2} + 20 \sqrt{10} + 20 = 30 + 30\sqrt{2} + 20 \sqrt{10} + 20\).

Step 4: Combine like terms.

Combine like terms to get the final simplified result: \(50 + 30\sqrt{2} + 20 \sqrt{10}\).

Key Concepts

Multiplying Radicals

When multiplying radicals, it's important to remember a few key principles that make the process smoother. Radicals, especially square roots, follow specific algebraic rules. One such rule is that the product of two radicals can be simplified using the identity:

• \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\).
• If both radicals are the square root of the same number, then \(\sqrt{a} \cdot \sqrt{a} = a\).

Applying these, let's examine a practical example with our expression:
\((3 \sqrt{5} + 2 \sqrt{10})(2 \sqrt{5} + \sqrt{10})\). First, you multiply across the terms like a binomial expression:

• \(3 \sqrt{5} \cdot 2 \sqrt{5} = 6(\sqrt{5 \cdot 5}) = 30\)
• \(3 \sqrt{5} \cdot \sqrt{10} = 3(\sqrt{5 \cdot 10}) = 3 \sqrt{50}\)
• \(2 \sqrt{10} \cdot 2 \sqrt{5} = 4(\sqrt{10 \cdot 5}) = 4 \sqrt{50}\)
• \(2 \sqrt{10} \cdot \sqrt{10} = 2(\sqrt{10 \cdot 10}) = 20\)

Each multiplication here uses the rules outlined above, which help transform and manage the radicals step by step. Simplifying radicals comes next after multiplying.

Simplifying Radicals

Simplifying radicals involves breaking down complex radical expressions into simpler forms. This part of arithmetic often confuses students until they grasp the foundational concept of prime factorization. The aim is to rewrite the number under the radical in terms of its perfect squares to extract it with ease.
Consider \(\sqrt{50}\) from the expansion step we just did. Decomposing 50 gives us \(5^2 \times 2\). Since 25 is a perfect square, you can simplify it as follows:

• \(\sqrt{50} = \sqrt{25 \cdot 2} = \sqrt{25} \cdot \sqrt{2} = 5\sqrt{2}\)

Now you integrate this simplification into our expression:
\(30 + 3 \cdot 5 \sqrt{2} + 20 \sqrt{10} + 20\) simplifies further to:
\(50 + 15\sqrt{2} + 20\sqrt{10}\).
Grouping like terms and reducing the radicals adjusts the expression to a cleaner, concise format.

Binomial Expansion

The binomial expansion involves multiplying two binomials to produce a quadratic or higher degree polynomial. It is a method used to expand expressions that raise one term within two sets of parentheses. A classic algebraic technique, binomial expansion extends expressions literally, using the distributive property.
For example, in \((3 \sqrt{5} + 2 \sqrt{10})(2 \sqrt{5} + \sqrt{10})\), reorder each component to cover all possible pairwise combinations:

• \((3 \sqrt{5}) \cdot (2 \sqrt{5})\)
• \((3 \sqrt{5}) \cdot (\sqrt{10})\)
• \((2 \sqrt{10}) \cdot (2 \sqrt{5})\)
• \((2 \sqrt{10}) \cdot (\sqrt{10})\)

This step-by-step distribution, often termed "FOIL" (First, Outer, Inner, Last), ensures that each term is calculated. Post-expansion, it becomes important to simplify your results, which involves the steps mentioned in multiplying and simplifying radicals. Proper understanding of binomial expansion opens the door to more elegant mathematical expression simplification and ultimately makes complex problems simpler to tackle.