Problem 87 - Intermediate Algebra

Question

a. Add: \(\sqrt{3}+\sqrt{3}\) b. Multiply: \(\sqrt{3} \cdot \sqrt{3}\) c. Describe the differences in parts (a) and (b).

Step-by-Step Solution

Answer

(a) The sum is \( 2\sqrt{3} \); (b) the product is 3. Addition involves combining like terms, while multiplication simplifies to a whole number.

Step 1: Understanding Addition of Radicals

To add radicals, the radicands (the numbers inside the square root) must be the same. In part (a), we have \( \sqrt{3} + \sqrt{3} \), and since the radicands are the same, we can add the coefficients (which are 1). Thus, the expression becomes \( 2 \times \sqrt{3} \) or simply \( 2\sqrt{3} \).

Step 2: Understanding Multiplication of Radicals

To multiply radicals, you can multiply the radicands directly. In part (b), we have \( \sqrt{3} \cdot \sqrt{3} \). Multiply the radicands to get \( 3 \), since \( \sqrt{3} \times \sqrt{3} = \sqrt{3 \times 3} = \sqrt{9} = 3 \).

Step 3: Comparing Addition and Multiplication of Radicals

In part (a), we are adding two radicals which results in \( 2\sqrt{3} \), combining like terms. In part (b), we multiply the radicals giving a whole number, \( 3 \), because \( \sqrt{3} \cdot \sqrt{3} \) converts to \( 3 \). The operations yield different types of results: one is a simplified radical expression and the other is a rational number.

Key Concepts

Addition of Radicals

Adding radicals is comparable to adding like terms in algebra. You can only directly add two radical terms if they have the same radicand. The radicand is the number inside the radical sign. For example, in the expression \( \sqrt{3} + \sqrt{3} \), both radicals have the same radicand, which is 3. Thus, you simply add the coefficients of the radicals.
In this case, the coefficients are 1 (since \( \sqrt{3} \) can be viewed as \( 1\cdot \sqrt{3} \)).

• First, check if the radicands are the same.
• If they are, add the coefficients directly.
• Write the result as a new coefficient in front of the radical.

So \( \sqrt{3} + \sqrt{3} = 2\cdot \sqrt{3} \), or simply \( 2\sqrt{3} \). This is what happens when the radicands match, making it easy to combine them.

Multiplication of Radicals

Multiplying radicals is often more straightforward than addition. When you multiply radicals, you do not need the radicands to be the same. Instead, you multiply the radicands together. For example, when multiplying \( \sqrt{3} \cdot \sqrt{3} \), you multiply the radicands 3 and 3.
This results in the expression \( \sqrt{3 \times 3} \).

• Perform multiplication of the radicands first.
• Simplify the expression inside the square root if possible.
• Convert into a whole number if the result is a perfect square.

Since \( 3 \times 3 = 9 \), and \( \sqrt{9} \) simplifies to 3, the expression \( \sqrt{3} \cdot \sqrt{3} = 3 \). Multiplication here transforms the radical to a rational number.

Simplifying Radicals

Simplifying radicals is a key skill needed for both adding and multiplying these expressions. Simplification involves breaking down the radicand into its simplest form. For addition, simplification allows you to see if the radicands can actually be added. In multiplication, simplification often comes at the end of the process.
Here's what you need to know:

• Factor the number inside the square root into its prime factors.
• Look for pairs of factors - these can be taken outside the square root.
• Write the simplified radical with any pairs as coefficients.

For example, to simplify \( \sqrt{12} \), we factor 12 into \( 2 \times 2 \times 3 \). The pair of twos can be taken outside, resulting in \( 2\sqrt{3} \). Simplification refines radical expressions, making computations like addition and multiplication more manageable.