Problem 18 - Algebra 1: Concepts and Skills

Question

Determine whether the radical expression is in simplest form. Explain. $$ 3 \sqrt{20} $$

Step-by-Step Solution

Answer

The simplified form of \( 3 \sqrt{20} \) is \( 6 \sqrt{5} \). Therefore, the given radical expression was not in its simplest form.

Step 1: Understanding the Problem

The given expression is \( 3 \sqrt{20} \). The goal is to simplify this expression, if possible.

Step 2: Prime Factorization of 20

Write 20 as the product of prime factors. This can be done by dividing it by prime numbers until all that is left are prime numbers. 20 divided by 2 gives 10, and 10 divided by 2 gives 5. So, the prime factorization of 20 is \( 2 * 2 * 5 \) or \( 2^2 * 5 \).

Step 3: Simplifying the Square Root

Simplify the square root using the rule \( \sqrt{a^2} = a \). Here, \( a = 2 \), so \( \sqrt{2^2} = 2 \). Now, the number inside the square root can be rewritten as \( \sqrt{2^2 * 5} = \sqrt{2^2} * \sqrt{5} = 2 * \sqrt{5} \).

Step 4: Multiplication

Finally, multiply the number outside the square root with the number obtained from the simplification process. Hence, \( 3 * 2 \sqrt{5} = 6 \sqrt{5} \).

Key Concepts

Prime Factorization

Prime factorization is the process of breaking down a composite number into its prime factors, which are numbers that are only divisible by 1 and themselves. Prime factors are the building blocks of all numbers, and this concept is crucial when simplifying radical expressions.

For instance, when we are given a number like 20 and we need to simplify a square root containing this number, we start by identifying its prime factors. To do this, we divide 20 by the smallest prime number, which is 2, and we get 10. We repeat this process until we end up with prime numbers: 20 is divided by 2 to get 10, then 10 is divided by 2 to get 5, which is a prime number. So, the prime factorization of 20 is represented as the product of its prime factors, which is written as 2 x 2 x 5 or in exponential form as 2^2 x 5.

Radical Simplification Rules

Radical simplification rules are the guidelines used to rewrite expressions involving square roots (or higher roots) in their simplest form. One of the fundamental rules is that the square root of a perfect square—a number that is the square of an integer—is just that integer.

Consider the expression \( \sqrt{a^2} = a \). This rule indicates that if we can express the number inside the square root as a perfect square multiplied by other factors, we can take the square root of the perfect square out of the radical, leaving the non-perfect square factors still under the radical.

Example of a Simplification:

If we apply this to \( \sqrt{2^2 \cdot 5} \), we recognize \( 2^2 \) as a perfect square. Thus, according to the rule, \( \sqrt{2^2 \cdot 5} \ = \sqrt{2^2} \cdot \sqrt{5} = 2 \cdot \sqrt{5} \). It's important to note that this process only works with multiplication inside the radical.

Square Roots

A square root is a mathematical operation that asks the question: 'What number, multiplied by itself, will give me the original number?' It's denoted by the radical sign \( \sqrt{} \). For example, since 4 is 2 times 2, the square root of 4 is 2, which we write as \( \sqrt{4} = 2 \).

However, not all square roots are so straightforward. Square roots of non-perfect squares, such as \( \sqrt{5} \), \( \sqrt{3} \) or \( \sqrt{2} \) do not yield neat integers. These roots are known as irrational numbers and can only be approximated when expressed as decimals. Simplifying expressions with square roots often involves finding the largest perfect square factor of a number while leaving the rest inside the square root.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations (like addition and multiplication) that represent a specific value or set of values. Simplifying algebraic expressions is a common process in algebra that involves reducing an expression to its easiest form by following mathematical operations and rules.

When dealing with expressions containing radicals, algebra comes into play in visible ways. After simplifying the radical component of an expression, typical algebraic operations such as multiplication may be needed to obtain the final simplified form. For instance, in the expression \( 3 \sqrt{20} \), after determining that \( \sqrt{20} \ = \sqrt{2^2 \cdot 5} = 2 \cdot \sqrt{5} \), we use algebraic multiplication to combine the numerical coefficient outside the radical with the result of the radical simplification, giving us \( 3 \cdot 2 \cdot \sqrt{5} = 6 \sqrt{5} \).