Problem 44 - Algebra and Trigonometry

Question

\(39-48=\) Simplify the expression. $$ \sqrt{8}+\sqrt{50} $$

Step-by-Step Solution

Answer

The expression simplifies to \( 7\sqrt{2} \).

Step 1: Simplify each square root

First, simplify each square root individually. Start with \( \sqrt{8} \). Notice that 8 can be factored into \( 4 \times 2 \), where 4 is a perfect square. Thus, \( \sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2 \sqrt{2} \).

Step 2: Simplify the second square root

Now, simplify \( \sqrt{50} \). The number 50 can be factored into \( 25 \times 2 \), where 25 is a perfect square. Therefore, \( \sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5 \sqrt{2} \).

Step 3: Combine simplified expressions

Now that we have simplified both square roots, replace them in the original expression: \( 2\sqrt{2} + 5\sqrt{2} \). Since both terms are like terms (they both include \( \sqrt{2} \)), we can add them together: \( (2 + 5)\sqrt{2} = 7\sqrt{2} \).

Key Concepts

Square Roots

Square roots are mathematical expressions that represent a value which, when multiplied by itself, equals the number under the root symbol. The symbol for square root is \( \sqrt{} \). For example, the square root of 9 is 3 because \( 3 \times 3 = 9 \). Understanding square roots is fundamental to simplifying expressions that contain them. When you encounter square roots in expressions, your goal is to simplify them so calculations become easier. This often involves factoring the number under the square root into its components, some of which are perfect squares.

Perfect Squares

Perfect squares are numbers that have integer square roots. For example, 4, 9, 16, and 25 are all perfect squares because their square roots are 2, 3, 4, and 5, respectively. Recognizing perfect squares is key to simplifying square root expressions. For instance, in an expression like \( \sqrt{50} \), you would notice that 50 can be factored into 25 and 2, where 25 is a perfect square. So, \( \sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5 \sqrt{2} \). Spotting these perfect squares quickly will help you break down square roots effectively and streamline your calculations.

Factoring

Factoring means breaking down a number into its multiplicative components, often to simplify mathematical expressions or solve equations. This is particularly useful when dealing with square roots as it helps identify perfect squares within the number. For example, if you want to simplify \( \sqrt{8} \), you begin by factoring 8 into 4 and 2. Since 4 is a perfect square, you can simplify \( \sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2} \). Factoring is not only helpful for understanding the structure of numbers but also essential for simplifying and solving problems that involve roots and other complex mathematical operations. Remember, when factoring, always look for perfect squares first, as they simplify the process significantly.