Problem 31 - College Algebra

Question

In Exercises \(27-38,\) add or subtract terms whenever possible. $$\sqrt{8}+3 \sqrt{2}$$

Step-by-Step Solution

Answer

The simplified form of the expression is \(5\sqrt{2}\).

Step 1: Simplify the root

Simplify \(\sqrt{8}\) to be \(2\sqrt{2}\). This is done by recognizing that 8 can be factored into 4 and 2, and square root of 4 is 2.

Step 2: Add like terms

Now, add together \(2\sqrt{2}\) and \(3\sqrt{2}\) to give \(5\sqrt{2}\). This step is possible because both terms are like terms, meaning they share the same radical part \(\sqrt{2}\).

Key Concepts

Radical Expressions

A radical expression is a mathematical phrase that includes a radical symbol, which is represented by \. The most common type of radical is the square root, but there can also be cube roots, fourth roots, and so on. Simplifying radical expressions often involves breaking down the number inside the radical (the radicand) into its prime factors and looking for perfect squares (or cubes, etc.) that can be taken out from under the radical sign.

For instance, in our exercise, \(\sqrt{8}+3 \sqrt{2}\), the number 8 inside the radical can be factored into 4 and 2. Since 4 is a perfect square, we take the square root of 4, which is 2, and place it outside the radical, reducing \(\sqrt{8}\) to \(2\sqrt{2}\). This process of simplification makes the expression easier to work with and sets the stage for further operations like addition or subtraction.

Like Terms

In algebra, 'like terms' are terms that have the same variables raised to the same power. When dealing with square roots, like terms refer to radical expressions that have the same radicand. For example, \(2\sqrt{2}\) and \(3\sqrt{2}\) are like terms because they both have the same radical part \(\sqrt{2}\).

In contrast, \(\sqrt{2}\) and \(\sqrt{3}\) are not like terms because they have different radicands. Like terms can be added or subtracted from each other, much like regular numerical terms. This concept is fundamental when simplifying expressions or solving equations.

Adding Radicals

Adding radicals works similarly to adding variables; you can only combine like terms. When you have two or more radical expressions that share the same radicand, you can simply add (or subtract) the coefficients, leaving the radical part unchanged.

Following our problem, after simplifying \(\sqrt{8}\) to \(2\sqrt{2}\), we add it to \(3\sqrt{2}\) because they are like terms. The sum becomes \(2+3\sqrt{2}\), which simplifies to \(5\sqrt{2}\). It is essential to look for opportunities to combine radicals throughout algebra because it helps in finding the simplest form of expressions.

Square Roots

A square root is a number that, when multiplied by itself, gives the original number. It is represented by the radical symbol with an index of 2 (implied when not written), for instance, \(\sqrt{9} = 3\) because \(3 \times 3 = 9\).

Understanding how to work with square roots is integral to simplifying radicals. For perfect squares like 4, 9, 16, and so forth, the square roots are whole numbers. However, not all square roots are neat and tidy. For example, \(\sqrt{2}\) is an irrational number, but it still follows the same rules for simplification and combination with like terms. Square roots play a significant role in geometry, algebra, and even higher levels of mathematics.