Problem 214 - Elementary Algebra

Question

In the following exercises, simplify. $$ \sqrt{150}+4 \sqrt{6} $$

Step-by-Step Solution

Answer

The simplified form is \( 9\sqrt{6} \).

Step 1 - Factor Inside the Square Root

First, factor the number inside the square root to find perfect squares. Notice that 150 can be factored as follows: \[ 150 = 25 \times 6 \].

Step 2 - Simplify the Square Root

Rewrite the square root expression to separate the perfect square: \[ \ \rightarrow \sqrt{150} = \sqrt{25 \times 6} \ = \sqrt{25} \times \sqrt{6} \ = 5\sqrt{6} \].

Step 3 - Combine Like Terms

Combine terms involving \(\sqrt{6}\). Rewrite the expression as: \[ \ \rightarrow 5\sqrt{6} + 4\sqrt{6} = (5 + 4)\sqrt{6} = 9\sqrt{6} \].

Key Concepts

Factoring

Before simplifying square roots, it's helpful to understand the concept of factoring. Factoring involves breaking down a number into smaller numbers that, when multiplied together, produce the original number. For example, 150 can be broken down into 25 and 6 because 25 multiplied by 6 equals 150. Factoring helps us identify perfect squares within a number, making it easier to simplify square roots. To factor a number, look for pairs of numbers that multiply to give the original number. List down these factors and check for any perfect squares amongst them.

Perfect Squares

Perfect squares are numbers that can be expressed as the product of an integer multiplied by itself. For example, 25 is a perfect square because it is 5 times 5. Recognizing perfect squares is essential when simplifying square roots. In the exercise, we factored 150 into 25 and 6 and realized that 25 is a perfect square. This lets us simplify \(\text{\sqrt{150}}\) into \(\text{5\sqrt{6}}\) because \(\text{\sqrt{25} = 5}\). Identifying and using perfect squares helps reduce complex square roots into simpler forms.

Combining Like Terms

Combining like terms means adding or subtracting terms that have the same variable and exponent. In the context of the given exercise, like terms are those involving \(\text{\sqrt{6}}\). Once we've simplified \(\text{\sqrt{150}}\) to \(\text{5\sqrt{6}}\), we add it to \(\text{4\sqrt{6}}\). Since both terms have the same radical part \(\text{(\sqrt{6})}\), we combine them by adding their coefficients: 5 and 4. This gives us \((5 + 4)\sqrt{6} = 9\sqrt{6}\). Always combine like terms to simplify expressions effectively.