Problem 62 - A Graphical Approach to College Algebra

Question

Simplify each expression, assuming that all variables represent nonnegative real numbers. $$2 \sqrt{5}-3 \sqrt{20}+2 \sqrt{45}$$

Step-by-Step Solution

Answer

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

Step 1: Understand the Problem

We are given the expression \(2 \sqrt{5}-3 \sqrt{20}+2 \sqrt{45}\) and need to simplify it. To do this, we will simplify each square root term individually to help combine them, if possible.

Step 2: Simplify Each Square Root Term

Start by simplifying each square root term in the expression:1. \( \sqrt{5} \) is already in its simplest form.2. For \( \sqrt{20} \), we factor 20 as \( 4 \times 5 \), so \( \sqrt{20} = \sqrt{4 \cdot 5} = \sqrt{4} \cdot \sqrt{5} = 2\sqrt{5} \).3. For \( \sqrt{45} \), we factor 45 as \( 9 \times 5 \), so \( \sqrt{45} = \sqrt{9 \cdot 5} = \sqrt{9} \cdot \sqrt{5} = 3\sqrt{5} \).

Step 3: Substitute Simplified Terms

Substitute the simplified terms back into the original expression:\(2 \sqrt{5} - 3 (2\sqrt{5}) + 2 (3\sqrt{5}) = 2 \sqrt{5} - 6\sqrt{5} + 6\sqrt{5}\).

Step 4: Combine Like Terms

Combine the like terms (those that include \( \sqrt{5} \)) in the expression:\((2 \sqrt{5} - 6 \sqrt{5} + 6 \sqrt{5}) = (2 - 6 + 6) \sqrt{5} = 2\sqrt{5}\).

Step 5: Final Simplified Expression

The expression \(2 \sqrt{5} - 3 \sqrt{20} + 2 \sqrt{45}\) simplifies to \(2 \sqrt{5}\).

Key Concepts

Square Roots

Square roots are mathematical operations that help us find a number which, when multiplied by itself, gives the original number. For example, the square root of 25 is 5 because 5 multiplied by 5 equals 25. Not all numbers have a neat square root like 25 does. Some, like 20 or 45, require simplification.To simplify square roots, look for perfect squares—that is, numbers like 4, 9, 16, etc., which have whole numbers as their square roots. For instance:

• To simplify \( \sqrt{20} \), recognize that 20 can be factored into 4 and 5, and since 4 is \( 2^2 \), it becomes \( \sqrt{4} \cdot \sqrt{5} = 2\sqrt{5} \).
• For \( \sqrt{45} \), factor 45 into 9 and 5. Recognizing 9 as a perfect square \( 3^2 \), the expression becomes \( \sqrt{9} \cdot \sqrt{5} = 3\sqrt{5} \).

This way, you can simplify radicals by extracting their square root components before moving on to combine them.

Combining Like Terms

Combining like terms is an essential part of simplifying mathematical expressions, especially when dealing with square roots.When an expression involves terms that have the same radical part, such as \( 2\sqrt{5} \) and \( 3\sqrt{5} \), you can combine them just as you would with regular algebraic terms. The key is to focus on the coefficients (the numbers in front of the square roots). For instance, in the expression \(2 \sqrt{5} - 6\sqrt{5} + 6\sqrt{5} \), the radical part, \( \sqrt{5} \), remains unchanged, but the coefficients can be added or subtracted:

• Add or subtract the coefficients: \(2 - 6 + 6 = 2\).
• The simplified expression becomes \(2\sqrt{5}\).

This technique is similar to combining like terms in algebra, where you would group and simplify the terms based on their variables.

Factorization

Factorization is breaking down numbers into their prime components or other multiply-able elements. This skill helps simplify square roots and discover commonalities in different terms.When working with expressions involving radicals, factorization allows us to simplify the square roots more effectively:

• Take \( \sqrt{20} \) as an example. Factorizing 20 gives \( 4 \times 5 \). The \( \sqrt{4} = 2 \), which simplifies the term to \( 2\sqrt{5} \).
• For \( \sqrt{45} \), factor 45 into 9 and 5. Since \( \sqrt{9} = 3 \), it becomes \( 3\sqrt{5} \).

Through factorization, you can pull out the square roots of perfect squares from a factorized number, making it easier to simplify and combine like terms.