Problem 21 - College Algebra

Question

Simplify each expression. $$\sqrt{150}$$

Step-by-Step Solution

Answer

The simplified form of \( \sqrt{150} \) is \( 5 \sqrt{6} \).

Step 1: Prime Factorization

First, we need to find the prime factors of 150. To do this, we start by dividing 150 by the smallest prime number, 2: 150 ÷ 2 = 75. Next, divide 75 by 3: 75 ÷ 3 = 25. Finally, divide 25 by 5: 25 ÷ 5 = 5, and we divide by 5 again: 5 ÷ 5 = 1. So, the prime factors of 150 are 2, 3, and 5².

Step 2: Express Under Square Root

Using the prime factors, we can express 150 as a product under the square root: \( \sqrt{150} = \sqrt{2 \times 3 \times 5^2} \).

Step 3: Simplify by Separating Perfect Square

Identify the perfect square factor from the expression under the radical sign:\( \sqrt{2 \times 3 \times 5^2} = \sqrt{(5^2) \times (2 \times 3)} \).Since 5² is a perfect square, it can be taken out of the radical:\( \sqrt{5^2} = 5 \).This simplifies the expression to:\( 5 \times \sqrt{6} \).

Step 4: Final Step: Write the Simplified Expression

Finally, write down the simplified form of the initial expression:\( \sqrt{150} = 5 \sqrt{6} \).

Key Concepts

Prime Factorization

Prime factorization is the process of breaking down a composite number into its prime factors. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. For example, the number 150 is not a prime number, so it can be expressed as a product of prime numbers.

To perform the prime factorization of 150, follow these steps:

• Start with the smallest prime number, which is 2, and check if it divides 150 evenly. Since 150 is even, 2 is a factor: 150 ÷ 2 = 75.
• Next, use 3, the next smallest prime number, to divide 75: 75 ÷ 3 = 25.
• Continue with the next smallest prime, which is 5, and divide 25: 25 ÷ 5 = 5.
• Finally, 5 divided by 5 equals 1, which completes the factorization: 5 ÷ 5 = 1.

Prime factors of 150 are found to be 2, 3, and 5², meaning you can express 150 as \( 2 \times 3 \times 5^2 \).

Knowing the prime factors helps in identifying perfect squares and simplifying radical expressions.

Perfect Squares

Perfect squares are numbers that are the result of squaring an integer. For example, 25 is a perfect square because it is \( 5^2 \). Identifying perfect squares is crucial when simplifying radical expressions because they allow us to take numbers out of the square root.

In the expression \( \sqrt{150} \), we can identify perfect squares from the prime factorization \( 2 \times 3 \times 5^2 \).

• The factor \( 5^2 \) is a perfect square.
• This means \( \sqrt{5^2} \) equals 5, which can be removed from the radical.

By rearranging these factors, you extract the perfect square component from the radical:

\( \sqrt{2 \times 3 \times 5^2} = \sqrt{5^2} \times \sqrt{2 \times 3} = 5 \times \sqrt{6} \).

Perfect squares simplify the computation and help express radical expressions in their simplest form.

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, etc. They often need simplification to make calculations easier. Simplifying radicals is done by identifying and extracting any perfect square factors.

For example, we simplify \( \sqrt{150} \) by using its prime factorization. After expressing 150 as \( \sqrt{2 \times 3 \times 5^2} \), we utilized the perfect square \( 5^2 \).

• The expression \( \sqrt{150} \) is simplified by taking \( 5^2 \) out of the radical, resulting in \( 5 \times \sqrt{6} \).
• The number 6 remains under the square root because it is not a perfect square.

Thus, \( \sqrt{150} = 5 \sqrt{6} \). This expression is simpler and more useful for further calculations or analyses. Understanding radical expressions and their simplification is essential in algebra and helps solve more complex mathematical problems with ease.