Problem 38 - College Algebra Essentials

Question

$$\sqrt{20}+6 \sqrt{5}$$

Step-by-Step Solution

Answer

\( 8 \sqrt{5} \)

Step 1: Simplify Square Root

First, simplify the square root of 20. This can be achieved by factoring out the perfect squares from \( \sqrt{20} \). The greatest perfect square factor of 20 is 4. So, \( \sqrt{20} \) can be written as \( \sqrt{4*5} \). Which simplifies further to \( 2 \sqrt{5} \).

Step 2: Combine Like Terms

Next combine the like terms. In this case, the like terms are \( 2 \sqrt{5} \) and \( 6 \sqrt{5} \). When combined, you get \( 8 \sqrt{5} \).

Key Concepts

Square Roots

Square roots are an essential mathematical concept often encountered when simplifying radical expressions. But what exactly is a square root? Simply put, the square root of a number is a value which, when multiplied by itself, gives the original number.
For example, the square root of 16 is 4 because \(4 \times 4 = 16\). More formally, the square root is represented by the radical symbol \(\sqrt{}\). Calculating the square root involves pinpointing that magical number under the radical.

Sometimes, numbers under the radical can be simplified by factoring out perfect squares. In our example, 20 was broken down to \(4 \times 5\), allowing for its square root to be expressed as \(\sqrt{4 \times 5} = \sqrt{4} \times \sqrt{5}\). Since 4 is a perfect square (\(2 \times 2 = 4\)), it can be simplified further to \(2 \sqrt{5}\).
This type of simplification helps when dealing with radical expressions in more complex maths.

Factoring

Factoring is a handy tool when working with radicals. It helps break down a number into its base components, simplifying the process of finding square roots. But what does factoring entail? Factoring means expressing a number as the product of its prime numbers or breaking it down to the simplest components.

For example, the number 20 can be written as \(4 \times 5\). 4, in this case, is a perfect square while 5 is a prime number. By recognizing the number 4, we can simplify the expression \(\sqrt{20}\) to make it easier to work with.

• Identify numbers easily factored into perfect squares like 4, 9, 16, 25, 36, etc.
• Factor the base number while aiming to pull out the greatest perfect square.
• The result will be a simpler radical expression.

Breaking down numbers to their simplest forms not only simplifies the square roots but also makes what was once complex into something more approachable.

Like Terms

When working with expressions, identifying and combining like terms is crucial for simplification. Like terms are simply terms in an expression that contain the same variables raised to the same power, with each term differentiated only by its coefficient.

In the expression \(2 \sqrt{5} + 6 \sqrt{5}\), both terms are like terms because they share the same radical part, \(\sqrt{5}\). They differ only in their coefficients, which are 2 and 6 respectively.
To simplify, combine these coefficients: \(2 + 6 = 8\). This results in the simplified expression \(8 \sqrt{5}\).

• Identify like terms by checking for the same variables or radicals.
• Combining them involves simply adding or subtracting their coefficients.
• This step brings clarity and simplicity to expressions.

Mastering the art of simplifying like terms is invaluable, particularly when tackling larger, more complex mathematical problems.