Problem 85 - Intermediate Algebra

Question

Simplify each expression, if possible. All variables represent positive real numbers. $$ \sqrt{300 x y} $$

Step-by-Step Solution

Answer

The expression simplifies to \(10 \sqrt{3xy}\).

Step 1: Prime Factorization

The first step is to find the prime factorization of the constant under the square root. The number 300 can be factored into prime numbers as follows: \[ 300 = 2^1 \times 3^1 \times 5^2 \times 2^1 = 2^2 \times 3^1 \times 5^2 \]This allows us to simplify the square root later.

Step 2: Simplify the Square Root

Next, we simplify the square root expression using the prime factors:\[\sqrt{300xy} = \sqrt{2^2 \times 3^1 \times 5^2 \times xy}\]We can separate the square root into distinct components:\[\sqrt{2^2} \times \sqrt{3} \times \sqrt{5^2} \times \sqrt{x} \times \sqrt{y}\]Where \( \sqrt{2^2} = 2 \) and \( \sqrt{5^2} = 5 \).

Step 3: Multiply Simplified Terms

Multiply the simplified terms obtained from the square root:\[2 \times 5 \times \sqrt{3} \times \sqrt{x} \times \sqrt{y} = 10 \sqrt{3xy}\]This is the simplified form of the original expression: \(10 \sqrt{3xy}\).

Key Concepts

Prime Factorization

Prime factorization is a method used to express numbers as a product of prime numbers. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves.

For example, to perform prime factorization on 300, you start by dividing by the smallest prime number:

• 300 is even, so divide by 2: \(300 \div 2 = 150\).
• Repeat with 150: divide by 2 again: \(150 \div 2 = 75\).
• 75 is not divisible by 2 but is divisible by 3: \(75 \div 3 = 25\).
• Finally, 25 is divisible by 5 twice: \(25 \div 5 = 5\) and \(5 \div 5 = 1\).

This shows that 300's prime factorization is \(2^2 \times 3^1 \times 5^2\). This breakdown helps simplify the square root, as it allows us to identify pairs of prime numbers that can be taken out of the root.

Square Root

The square root is a special mathematical function that computes a number which, when multiplied by itself, equals the given value. For instance, the square root of 9 is 3 because \(3 \times 3 = 9\).

In simplifying radical expressions like \(\sqrt{300xy}\), we first use prime factorization. Then, any time a pair of identical numbers (or letters) exists, one number is pulled outside the square root.

• In \(\sqrt{2^2}\), the square root simplifies to 2, since \(2 \times 2\) forms a pair.
• \(\sqrt{5^2} = 5\) is similar, as it also forms a pair.
• \(\sqrt{3}\), \(\sqrt{x}\), and \(\sqrt{y}\) have no pairs, so they remain inside the root.

This process enables us to express the expression efficiently as \(10 \sqrt{3xy}\). Remember, simplifying square roots makes solving and comprehending mathematical problems more manageable.

Positive Real Numbers

Positive real numbers are numbers that are greater than zero. They include integers, fractions, and irrational numbers such as decimals that do not have a terminating pattern. When working with square roots and radical expressions, it's important to consider that the variables involved are positive real numbers.

This consideration eliminates complexities associated with negative numbers under square roots, which frequently involve imaginary numbers. In the expression \(\sqrt{300xy}\), the variables \(x\) and \(y\) are defined to be positive real numbers. This definition allows us to safely assume that we are always dealing with positive values after simplification.

Understanding this concept is crucial because it ensures all mathematical operations, especially involving roots, yield results within the real number system without introducing complex numbers unintentionally. This clarity helps students focus on the techniques of simplification without complications.