Problem 13 - Algebra 2

Question

Simplify. $$ \sqrt{225} $$

Step-by-Step Solution

Answer

The simplified form of \( \sqrt{225} \) is 15.

Step 1: Understand the Problem

We need to simplify the expression \( \sqrt{225} \). This means we are looking for a number that, when multiplied by itself, gives 225.

Step 2: Break Down 225 into Factors

Consider the factors of 225 to find a perfect square within it. By performing prime factorization:225 is divisible by 3:225 ÷ 3 = 7575 is divisible by 3:75 ÷ 3 = 2525 is divisible by 5:25 ÷ 5 = 5Finally, 5 is divisible by 5:5 ÷ 5 = 1The prime factorization of 225 is \( 3^2 \times 5^2 \).

Step 3: Apply Square Root Property

From the factorization, \( 225 = 3^2 \times 5^2 \). We can use the property \( \sqrt{a^2} = a \) to simplify this:\( \sqrt{225} = \sqrt{3^2 \times 5^2} = \sqrt{3^2} \times \sqrt{5^2} = 3 \times 5 \).

Step 4: Simplify

Calculate the multiplication:\( 3 \times 5 = 15 \). Thus, \( \sqrt{225} = 15 \).

Key Concepts

Prime Factorization

Prime factorization is the process of expressing a number as the product of its prime factors. Prime factors are the building blocks of a number, meaning they are prime numbers that multiply together to form the original number. When simplifying square roots, prime factorization can help identify patterns or simplify calculations. To factor a number, continuously divide it by the smallest prime number until the result is 1.

For example, let's factor 225. This number divides evenly by the prime number 3:

• 225 ÷ 3 = 75

75 can again be divided by 3:

• 75 ÷ 3 = 25

25 is not divisible by 3, but it can be divided by the next smallest prime number, which is 5:

• 25 ÷ 5 = 5

And finally, 5 is divided by itself:

• 5 ÷ 5 = 1

Thus, the prime factorization of 225 is expressed as:

• \( 225 = 3^2 \times 5^2 \)

This breakdown helps simplify square roots by recognizing groups of two prime factors, leading to the next step of identifying perfect squares.

Perfect Squares

A perfect square is a number that is the product of an integer multiplied by itself. Recognizing perfect squares is handy when simplifying square roots because the square root of a perfect square is always an integer. When you spot numbers in a factorization that appear in pairs, they can be simplified as perfect squares.

Using the example of 225, whose prime factorization is \( 3^2 \times 5^2 \), both 3 and 5 appear squared:

• \( 3^2 \) is a perfect square because \( 3 \times 3 = 9 \) and \( \sqrt{9} = 3 \).
• \( 5^2 \) is also a perfect square because \( 5 \times 5 = 25 \) and \( \sqrt{25} = 5 \).

This understanding allows us to simplify the entire expression inside the square root by taking the square root of each perfect square individually. Therefore, recognizing perfect squares and their properties are essential skills for simplifying expressions involving square roots.

Square Root Property

The square root property enables us to simplify expressions involving square roots, especially when they contain perfect squares. The property states that for any non-negative number, \( \sqrt{a^2} = a \). This makes it straightforward to handle square roots of perfect squares.

To simplify \( \sqrt{225} \) using this property, first recognize the prime factorization: \( 225 = 3^2 \times 5^2 \). Using the square root property, take the square root of each grouped prime factor:

• \( \sqrt{3^2} = 3 \)
• \( \sqrt{5^2} = 5 \)

Once you have the square roots of the perfect squares, multiply them together:

• \( 3 \times 5 = 15 \)

Thus, \( \sqrt{225} = 15 \). This method makes it uncomplicated to handle square roots of numbers that are perfect squares, as you simply identify the squared numbers and apply the property to arrive at an integer result.