Problem 177 - Elementary Algebra

Question

In the following exercises, simplify. $$ \sqrt{50}+4 \sqrt{2} $$

Step-by-Step Solution

Answer

\(9 \sqrt{2}\)

Step 1 - Simplify the Square Root

Identify the factors of the number inside the square root. For \(\text{50}\), the factors are \(\text{25} \times \text{2}\). Write the square root of 50 as \(\text{\sqrt{25} \times \sqrt{2}}\).

Step 2 - Simplify the Perfect Square

Simplify \(\text{\sqrt{25}}\), which is \(\text{5}\). This gives us \(5 \sqrt{2}\).

Step 3 - Combine Like Terms

Combine \(5 \sqrt{2}\) and \(4 \sqrt{2}\) by adding their coefficients. The result is \((5 + 4) \sqrt{2} = 9 \sqrt{2}\).

Key Concepts

square root simplification

Simplifying square roots can make certain mathematical expressions easier to work with. The goal is to express the square root in its simplest form. To do this, identify factors within the number inside the square root that are perfect squares.
For example, in the expression \(\text{sqrt(50) + 4 * sqrt(2)}\), we focus on \(\text{sqrt(50)}\).
Break down the number under the square root into its factors: \(\text{50 = 25 * 2}\).
Since \(\text{25}\) is a perfect square, it simplifies the expression when factored out:
\(\text{sqrt(25 * 2) = sqrt(25) * sqrt(2)}\) and \(\text{sqrt(25) = 5}\).
This ensures \(\text{sqrt(50) = 5 * sqrt(2)}\), a much simpler form to handle.
Following these steps, we have brought \(\text{sqrt(50)}\) to its simplified form \(\text{5 * sqrt(2)}\).

perfect squares

Understanding perfect squares is a crucial part of simplifying square roots. A perfect square is a number that can be expressed as the square of an integer. Examples include \(\text{1, 4, 9, 16, 25, etc.}\)
In the example \(\text{sqrt(50)}\), we use the perfect square \(\text{25}\), which simplifies nicely.
To find perfect squares in your expressions, look for numbers whose square roots are integers:

• \(\text{4 (2^2)}\)
• \(\text{9 (3^2)}\)
• \(\text{16 (4^2)}\)
• \(\text{25 (5^2)}\)

Using perfect squares allows for breaking down square roots efficiently.
Always search for the largest perfect square factor to make simplification easier and your final expressions cleaner.

combining like terms

In algebra, combining like terms is essential to simplifying expressions. Like terms have identical variable parts. In sqrt simplifications, we often encounter terms with the same square root.
For example, in \(\text{5 sqrt(2) + 4 sqrt(2)}\), both terms are like terms because they share the \(\text{sqrt(2)}\) factor.
Combining them involves adding the coefficients.
This means we add \(\text{5}\) and \(\text{4}\), giving us \(\text{(5 + 4) sqrt(2) = 9 sqrt(2)}\).
This approach simplifies the expression to its most reduced form efficiently.
Knowing how to combine like terms helps make complex mathematical expressions cleaner and easier to understand. Always look for terms with identical factors when simplifying expressions.