Problem 35 - Algebra 2 and Trigonometry

Question

In \(3-38\) , write each radical in simplest radical form. Variables in the radicand of an even index are non-negative. Variables occurring in the denominator of a fraction are non-zero. $$ \sqrt{300 c} $$

Step-by-Step Solution

Answer

The simplest form is \(10\sqrt{3c}\).

Step 1: Factor the Number Under the Radical

First, decompose the number 300 into its prime factors. The number 300 can be expressed as \(300 = 2^2 \times 3 \times 5^2\).

Step 2: Apply the Square Root to the Factors

Rewrite the expression \(\sqrt{300c}\) using the factors: \(\sqrt{2^2 \times 3 \times 5^2 \times c}\). Separate into two square roots: \(\sqrt{2^2 \cdot 5^2} \cdot \sqrt{3c}\).

Step 3: Simplify the Perfect Squares

Since \(\sqrt{2^2} = 2\) and \(\sqrt{5^2} = 5\), the expression simplifies to \(2 \times 5 \times \sqrt{3c}\). Thus, \(10\sqrt{3c}\).

Key Concepts

Prime Factorization

Prime factorization is a technique used to decompose a number into a product of its prime numbers. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. This method is particularly useful when dealing with radicals because it helps identify perfect squares, which can be simplified.Let's consider the number 300. To find its prime factors, we break it down step by step:

• Start by dividing it by the smallest prime number, 2: 300 ÷ 2 = 150
• Next, divide 150 by 2 again: 150 ÷ 2 = 75
• Now that 75 is odd, we switch to the next smallest prime number, 3: 75 ÷ 3 = 25
• Finally, divide 25 by the prime number 5: 25 ÷ 5 = 5, and one last time, 5 ÷ 5 = 1

Now, 300 can be expressed as a product of prime numbers: \(300 = 2^2 \times 3 \times 5^2\). This factoring is essential for simplifying radicals efficiently.

Square Root Properties

Understanding square root properties is key to simplifying expressions involving radicals. The square root of a product can be separated into the product of the square roots. This is useful when dealing with prime factors:For example, suppose you need to simplify \(\sqrt{300c}\). Given that 300 is factored into \(2^2 \times 3 \times 5^2\), square root properties allow you to rewrite the expression:

• \(\sqrt{300c} = \sqrt{2^2 \cdot 3 \cdot 5^2 \cdot c}\)
• = \(\sqrt{2^2} \times \sqrt{5^2} \times \sqrt{3c}\)

These properties are advantageous because any perfect squares can easily "come out" of the radical. For example, \(\sqrt{2^2} = 2\) and \(\sqrt{5^2} = 5\), simplifying the calculation and breaking it down into easier steps.

Simplifying Radical Expressions

Simplifying radical expressions is the process where you systematically break down and simplify the terms inside a radical. After applying prime factorization and square root properties, simplifying a radical expression becomes straightforward.Let's simplify \(\sqrt{300c}\). With all the preparation work done:

• We have \(\sqrt{2^2 \times 3 \times 5^2 \times c}\)
• Apply properties to get \(\sqrt{2^2} \times \sqrt{5^2} \times \sqrt{3c}\)
• Simplify the perfect squares: \(2 \times 5 = 10\)

This makes the simplified expression \(10\sqrt{3c}\). Importantly, simplifying radical expressions can make complex problems manageable and reduce errors in calculations.