Problem 60 - Algebra 1

Question

Simplify the radical expression. $$\sqrt{20}$$

Step-by-Step Solution

Answer

The simplified form of \(\sqrt{20}\) is \(2\sqrt{5}\).

Step 1: Find the Prime Factorization of the Number

Begin by finding the prime factorization of the number under the square root. In this case, factorize 20 into primes: \(20 = 2 \times 2 \times 5\) or \(2^2 \times 5^1\).

Step 2: Identify Pairs of Primes and Simplify

With the prime factorization, we can simplify. In this case, \(2^2\) can be taken out of the square root as 2 since it's a pair (any pair of identical numbers can be taken out of the square root). Thus, the expression becomes \(2\sqrt{5}\).

Step 3: Write the Simplified Expression

The simplified expression is \(2\sqrt{5}\).

Key Concepts

Prime Factorization

Prime factorization is a technique used for breaking down a number into a product of its prime numbers. Prime numbers are numbers greater than 1 that can only be divided by 1 and themselves, like 2, 3, 5, 7, and so on.
To factorize a number like 20, start by dividing it by the smallest prime number, 2. Hence, 20 divided by 2 gives 10. Do it again: 10 divided by 2 gives 5. Finally, 5 is a prime number by itself, hence you stop here.

You get:

• 20 = 2 × 2 × 5

Using prime factorization is crucial when simplifying radicals because it helps identify pairs of prime factors, which can be extracted from the radical for a more simplified expression.

Square Root

A square root refers to a number that, when multiplied by itself, gives the original number. The symbol \(\sqrt{\cdot}\) is used to denote the square root.
For example, \(\sqrt{16} = 4\) because \(4 \times 4 = 16\).
In our case with the expression \(\sqrt{20}\), we start by expressing 20 as \(2^2 \times 5\) from our prime factorization. Here, \(2^2\) indicates a perfect square under the root.

We can extract 2 from \(\sqrt{20}\) as it forms a pair:

• So \(\sqrt{20} = \sqrt{2^2 \times 5} = 2\sqrt{5}\)

This process ensures that you simplify the radical to make it more readable and easy to work with in equations or further calculations.

Algebra 1

Algebra 1 is a foundational course focusing on the basics of algebra, introducing you to expressions, equations, and various operations. Understanding these principles is crucial for progressing in mathematics.
Simplifying radicals like \(\sqrt{20}\) often appears in Algebra 1. This exercise helps you learn how to manipulate numbers and expressions.

The steps involve:

• Finding prime factors.
• Identifying and extracting pairs from under the square root.
• Simplifying the expression in terms of its simplest radical form.

Practicing these exercises enhances your algebraic thinking, making you comfortable handling more complex algebraic problems down the road.