Problem 94 - College Algebra Essentials

Question

Will help you prepare for the material covered in the next section. $$\text { Simplify: } \sqrt{18}-\sqrt{8}$$

Step-by-Step Solution

Answer

\(\sqrt{2}\)

Step 1: Simplify the Square Roots

We simplify the square roots separately by trying to express the numbers as a product of a perfect square and another factor. \n\n For \(\sqrt{18}\), the largest perfect square that divides it is 9, so \(18 = 9 \cdot 2\). Thus, \(\sqrt{18}= \sqrt{9\cdot2}= \sqrt{9} \cdot \sqrt{2}= 3\sqrt{2}\). \n\n Similarly, for \(\sqrt{8}\), the largest perfect square that divides it is 4, so \(8 = 4 \cdot 2\). Thus, \(\sqrt{8}= \sqrt{4\cdot2}= \sqrt{4} \cdot \sqrt{2}= 2\sqrt{2}\).

Step 2: Substitute the Simplified Roots into the Original Expression

Substitute the simplified roots, \(3\sqrt{2}\) and \(2\sqrt{2}\), back into the original expression: \[(3\sqrt{2}) - (2\sqrt{2})\]

Step 3: Simplify the Expression

The terms are like terms because they have the same square roots. Subtract the numbers in front of the square roots, and leave the square roots as they are. So, the simplified expression is: \[ 3\sqrt{2} - 2\sqrt{2} = \sqrt{2} \]

Key Concepts

Algebraic Expressions

Algebraic expressions are a fundamental concept in mathematics. They are combinations of variables, numbers, and operations like addition or subtraction. In the given problem, \(\sqrt{18} - \sqrt{8}\), we have an algebraic expression that involves square roots. Each part of the expression can be simplified to make the entire expression easier to work with.

Understanding and simplifying algebraic expressions is a crucial skill. Simplification can make expressions easier to compute, compare, and understand. This process involves recognizing parts of the expression that are alike and combining them.

In the given example, both terms are in the form of square roots. To solve the problem, we simplified these square roots first before dealing with the subtraction. This approach helps in breaking down complex expressions into manageable parts.

Square Roots

Square roots are all about finding a number that, when multiplied by itself, gives you the original number. The symbol \(\sqrt{}\) is commonly used to represent square roots. In this exercise, the expression involves the square roots \(\sqrt{18}\) and \(\sqrt{8}\). Here's how we handled them:

• For \(\sqrt{18}\): We split 18 into 9 and 2, because 9 is a perfect square (since \(3 \times 3 = 9\)). Thus, \(\sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2}\).
• For \(\sqrt{8}\): We write 8 as \(4 \times 2\), where 4 is a perfect square (\(2 \times 2 = 4\)). Hence, \(\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2}\).

Recognizing perfect squares helps to simplify square roots quickly. This makes algebraic operations smoother in expressions involving radicals.

Perfect Squares

Perfect squares are numbers that are the square of an integer. For example, 1, 4, 9, 16, and so on are perfect squares. They are crucial in simplifying square roots because any number that is a square of an integer is usually easier to handle in mathematical expressions.

In the original problem, identifying the perfect squares within 18 and 8 helps break them down effectively:

• For 18, 9 is a perfect square that divides 18 (since \(9 \times 2 = 18\)). So, \(9 = 3^2\).
• For 8, 4 is a perfect square that divides 8 (\(4 \times 2 = 8\)). So, \(4 = 2^2\).

Recognizing and using perfect squares empowers the simplification process of radicals. This often involves expressing a larger number as a product of its largest perfect square factor and another factor, making it easier to simplify complex expressions.