Problem 88 - Algebra A Combined Function

Question

a. Add: \(2 \sqrt{5}+\sqrt{5}\) b. Multiply: \(2 \sqrt{5} \cdot \sqrt{5}\) c. Describe the differences in parts (a) and (b).

Step-by-Step Solution

Answer

a. \(3 \sqrt{5}\); b. 10; c. Addition simplifies terms whereas multiplication changes the form.

Step 1: Understand Addition of Like Terms

In the expression \(2 \sqrt{5} + \sqrt{5}\), we are asked to add like terms. Like terms are terms that have the same radical part, in this case, \(\sqrt{5}\).

Step 2: Combine Like Terms

When adding \(2 \sqrt{5} + \sqrt{5}\), combine the coefficients of like terms. The expression can be rewritten as \((2+1) \sqrt{5} = 3 \sqrt{5}\). So, \(2 \sqrt{5} + \sqrt{5} = 3 \sqrt{5}\).

Step 3: Understand Multiplication of Radicals

For the expression \(2 \sqrt{5} \cdot \sqrt{5}\), we multiply the coefficients and the radicals separately.

Step 4: Multiply Coefficients and Radicals

Multiply the coefficients: \(2 \times 1 = 2\). Then, multiply the radicals: \(\sqrt{5} \times \sqrt{5} = \sqrt{5^2} = 5\). Therefore, the expression becomes \(2 \times 5 = 10\).

Step 5: Understand Differences Between Addition and Multiplication

In part (a), we combined like terms by adding their coefficients to get a single term. In part (b), we multiplied the entire term, involving both coefficient and radical, resulting in a non-radical number.

Key Concepts

Addition of Radicals

Adding radicals is similar to adding regular numbers, but only if they are 'like terms'. Like terms in radicals are those that have the same radicand (the number under the square root). For example, in the expression \(2 \sqrt{5} + \sqrt{5}\), both terms are like terms because they contain the same radicand \(\sqrt{5}\).
To add them, you only add their coefficients, which are the numbers in front of the square roots.

• Start with the expression: \(2 \sqrt{5} + \sqrt{5}\).
• Recognize that both terms are like terms because they have the same radical part \(\sqrt{5}\).
• Add their coefficients: \(2 + 1 = 3\).

Thus, the sum becomes \(3 \sqrt{5}\). It's crucial to note that you can only combine radicals in this manner if they are like terms. Different radicands, such as \(\sqrt{5}\) and \(\sqrt{7}\), cannot be added or directly combined.

Multiplication of Radicals

Multiplying radicals involves multiplying both the coefficients and the radicands separately. In multiplication, unlike addition, the terms do not need to be like terms.
Here's how multiplication of radicals is typically handled:

• Take the expression \(2 \sqrt{5} \cdot \sqrt{5}\).
• First, multiply the coefficients of the terms, i.e., \(2 \times 1 = 2\).
• Then, find the product of the radicands: \(\sqrt{5} \times \sqrt{5} = \sqrt{5^2} = 5\).

Hence, the overall product is \(2 \times 5 = 10\). The result shows how multiplication is different from addition because when you multiply radicals, any radical squared simplifies to a whole number. This is key in transforming an expression from a radical to a non-radical outcome.

Like Terms in Algebra

Understanding like terms is fundamental in simplifying algebraic expressions. Like terms are terms that have exactly the same variables or radicals raised to the same power. In the case of radicals, the concept extends to having the same radicand.
In the expression \(2 \sqrt{5} + \sqrt{5}\), the terms are considered like terms because:

• Both terms contain the same radical \(\sqrt{5}\).
• The coefficients can be combined directly: \(2 + 1 = 3\).

When you're dealing with algebraic expressions, always search for like terms to simplify.
Grouping like terms helps condense expressions into simpler terms, paving the way for clearer calculations and solutions. Remember, in terms of multiplication, this requirement of similarity isn't necessary. However, in addition and subtraction, recognizing like terms is crucial as they allow you to combine and simply the expression correctly.