Problem 26 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$ \sqrt{63} $$

Step-by-Step Solution

Answer

The square root of 63 is simplified to \(3 \sqrt{7}\).

Step 1: Identify Prime Factors

First identify the prime factors of 63. This can be done by using a factor tree. Here, 63 can be broken down to 3 x 3 x 7, which are all prime numbers.

Step 2: Simplify the Square Root

Next look for pairs of similar numbers under the root. Here, there are two 3s. One 3 can be taken out from the root. So, \( \sqrt{63} \) simplifies to \(3 \sqrt{7}\).

Key Concepts

Prime Factorization

Prime factorization is like finding the building blocks of a number using prime numbers. A prime number is a number greater than one that has no divisors other than 1 and itself. For example, 2, 3, 5, and 7 are all prime numbers because they can't be divided evenly by any number except for 1 and themselves.

To find the prime factors of 63, we start by dividing by the smallest prime number possible, which is 3.

• 63 divided by 3 is 21.
• Repeat the process with 21: dividing by 3 gives us 7 (since 21 / 3 = 7).
• Now, observe that 7 is a prime number itself.

Thus, by using a factor tree or similar method, 63 can be written as a product of prime numbers: 3 x 3 x 7. Each branch leads to a prime, showing how 63 breaks down into its simplest components.

Square Root Simplification

Square root simplification is a useful technique to make calculations easier while maintaining accuracy. It involves expressing a square root in its simplest form often by pulling out perfect squares from under the radical symbol.

When you break down a number into its prime factors, you can look for pairs of like factors. For instance, in the prime factorization of 63, which is 3 x 3 x 7, there is a pair of 3s.

• A pair of like factors inside the square root can be moved outside as a single number.
• This is because \(\sqrt{a \times a} = a\).

Applying this to \(\sqrt{63}\), we can take one 3 out of the square root, resulting in \(3 \sqrt{7}\). Now the square root is in a simplified form, making it easier to work with, especially in further calculations or algebraic expressions.

Radical Expressions

Radical expressions often involve square roots, cube roots, and other roots, and knowing how to handle them is key in math. A radical symbol \(\sqrt{}\) is used to denote the square root of a number, which basically asks, "What number multiplied by itself gives us this number?"

Simplifying radical expressions like \(\sqrt{63}\) involves a few simple strategies:

• Find the prime factorization of the number inside the radical.
• Look for pairs of identical factors, which can be pulled out from under the radical as one factor.
• Express the simplified radical, making calculations easier and clearer.

In the expression \(\sqrt{63}\), recognizing and utilizing like pairs (like the 3s) helps turn a complex expression into the more manageable \(3\sqrt{7}\). Understanding radical expressions is crucial for working in algebra and beyond, making problem-solving more efficient.