Problem 22 - Algebra 2

Question

Simplify. $$ \sqrt{125} $$

Step-by-Step Solution

Answer

The simplified form of \( \sqrt{125} \) is \( 5\sqrt{5} \).

Step 1: Identify Prime Factorization

First, we need to find the prime factors of the number inside the square root, which is 125. Start by dividing 125 by the smallest prime number, 5: \( 125 \div 5 = 25 \). Now factor 25: \( 25 \div 5 = 5 \). Since 5 is a prime number, the prime factorization of 125 is \( 5^3 \).

Step 2: Apply Square Root to Prime Factorization

Use the property \( \sqrt{a^2} = a \) to simplify the square root. Since we have \( \sqrt{5^3} \), we can rewrite \( 5^3 \) as \( 5^2 \times 5 \). Now, apply the square root: \( \sqrt{5^2 \times 5} = \sqrt{5^2} \times \sqrt{5} = 5 \times \sqrt{5} \).

Step 3: Simplified Expression

The expression is now simplified. The square root of 125 simplifies to \( 5\sqrt{5} \).

Key Concepts

Prime Factorization

Prime factorization is a method used to break down a number into its basic building blocks, known as prime numbers. In our example, we are dealing with the number 125. Prime numbers are numbers greater than 1 that are only divisible by 1 and themselves. To find the prime factorization, start by identifying the smallest prime number that can divide 125. Here, that number is 5.

• Divide 125 by 5 to get 25.
• Repeat the process with 25; it is also divisible by 5, giving you 5.
• Finally, 5 is itself a prime number.

Thus, we list the prime factors of 125 as three 5s, or mathematically, as \( 5^3 \). This step is crucial because it sets the stage for simplifying square roots by revealing any perfect squares within the number.

Square Root Property

The square root property is fundamental when dealing with the square roots of numbers, especially those with prime factorization. The property we use most often is \( \sqrt{a^2} = a \), which helps in simplifying expressions. In simplifying \( \sqrt{125} \), you take the prime factorization \( 5^3 \) and express it in a way that reveals perfect squares.

• Rewrite \( 5^3 \) as \( 5^2 \times 5 \).
• Use the square root property to separate it as \( \sqrt{5^2} \times \sqrt{5} \).
• Since \( \sqrt{5^2} = 5 \), you get \( 5 \times \sqrt{5} \).

This property is key to reducing the square root to a simpler form and allows us to handle square roots more systematically by identifying perfect squares within the number's factors.

Simplification Process

The simplification process ties together prime factorization and the square root property to express complicated square roots in a simpler form. Start by recalling your prime factorization from earlier steps. For \( \sqrt{125} \), which we determined had a prime factorization of \( 5^3 \), the goal is to search for any squares that can be simplified.

• From the prime factorization \( 5^3 \), we rewrite it as \( 5^2 \times 5 \).
• Applying the square root property gives \( \sqrt{5^2} \times \sqrt{5} \).
• Simplify the expression to \( 5 \times \sqrt{5} \), which is a much neater form.

Ultimately, the simplification process transforms \( \sqrt{125} \) to \( 5\sqrt{5} \) by systematically applying the understanding of both prime factorization and the square root property, resulting in less complex expressions that are more intuitive to work with in further calculations.