Problem 47 - Elementary Algebra

Question

Simplify each expression by performing the indicated operation. $$ (\sqrt{2}+\sqrt{5})(\sqrt{2}+3 \sqrt{5}) $$

Step-by-Step Solution

Answer

Question: Simplify the following algebraic expression: $(\sqrt{2}+\sqrt{5})(\sqrt{2}+3\sqrt{5})$ Answer: The simplified algebraic expression is $17+4\sqrt{10}$.

Step 1: Rewrite and identify the terms

Rewrite the given expression and identify the terms. $$ (\sqrt{2}+\sqrt{5})(\sqrt{2}+3\sqrt{5}) $$ We have four terms in total: \(\sqrt{2}\), \(\sqrt{5}\), \(\sqrt{2}\), and \(3\sqrt{5}\)

Step 2: Use the distributive property

Multiply each term in the first parenthesis by each term in the second parenthesis. This is also known as using the distributive property. $$ (\sqrt{2}+\sqrt{5})(\sqrt{2}+3\sqrt{5}) = \sqrt{2}\cdot\sqrt{2}+\sqrt{2}\cdot3\sqrt{5}+\sqrt{5}\cdot\sqrt{2}+\sqrt{5}\cdot3\sqrt{5} $$

Step 3: Multiply the terms with radicals

Multiply the terms with radicals. $$ 2+3\sqrt{10}+\sqrt{10}+15 $$

Step 4: Combine like terms

Combine the constant terms and the terms with radicals. $$ 2+15+3\sqrt{10}+\sqrt{10} = 17+4\sqrt{10} $$ So, the simplified expression is: $$ 17+4\sqrt{10} $$

Key Concepts

Simplifying Radicals

Radicals are expressions that include the square root, cube root, or any higher-order root. When simplifying such expressions, it is important to combine any like terms. The purpose of simplifying is to write the expression in its simplest form, making it easier to read and work with. Let's particularly focus on square roots, as they are frequent in algebraic operations.

• Identify if the radicals can be simplified further by factoring any perfect squares inside the radicand (the number inside the root).
• If you are multiplying radicals, remember the rule: \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \). This means you can multiply the numbers inside the radicals and take a single root of the product.
• An example from our exercise is multiplying \( \sqrt{2} \cdot \sqrt{2} = 2 \).

The end goal is to express radicals in their simplest form, revealing any rational numbers or easier expressions. When dealing with radical expressions, consistency in method and thorough factoring are keys to properly simplifying the radicals.

Distributive Property

The distributive property is a fundamental rule used to simplify expressions, especially when dealing with multiple terms. It allows you to remove parentheses by distributing one term into every term inside another set of parentheses. This property especially comes in handy when multiplying expressions. To apply the distributive property:

• Identify the two sets of terms that need to be multiplied.
• Multiply each term in the first set by each term in the second set.
• Write down each resulting product separately.

For our expression, \( (\sqrt{2}+\sqrt{5})(\sqrt{2}+3\sqrt{5}) \), \( \sqrt{2} \cdot \sqrt{2} = 2 \) and \( \sqrt{2} \cdot 3\sqrt{5} = 3\sqrt{10} \) show how we distribute \( \sqrt{2} \) over both terms within the other set. It helps to solve complex, multi-term expressions through systematic term-by-term multiplication.

Combining Like Terms

When simplifying algebraic expressions, combining like terms helps to consolidate similar terms into single terms. This is the step where you tidy up your expression after multiplying or adding terms. Like terms refer to terms that contain the same variable part raised to the same power. In our case with radicals, like terms are those with the same radical part. Here's how to combine:

• Identify terms that have the same radical or variable part.
• Add or subtract their coefficients.

In our exercise, after distributing and simplifying, we were left with \( 2 + 3\sqrt{10} + \sqrt{10} + 15 \). The like terms \( 3\sqrt{10} \) and \( \sqrt{10} \) were combined into \( 4\sqrt{10} \), while the constants 2 and 15 combined into 17. By combining these like terms, we arrived at the final simplified expression: \( 17 + 4\sqrt{10} \).