Problem 41 - Algebra and Trigonometry

Question

\(39-48=\) Simplify the expression. $$ \sqrt{125}-\sqrt{45} $$

Step-by-Step Solution

Answer

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

Step 1: Simplify the Radical Expressions

Start by simplifying each radical expression individually. First, look at \(\sqrt{125}\). We know that \(125 = 25 \times 5\), and can thus rewrite it as \(\sqrt{25 \times 5}\). Because \(25\) is a perfect square, we can simplify further: \(\sqrt{25 \times 5} = \sqrt{25} \times \sqrt{5} = 5\sqrt{5}\).

Step 2: Simplify the Second Radical

Next, simplify \(\sqrt{45}\). Notice that \(45 = 9 \times 5\), so we can rewrite it as \(\sqrt{9 \times 5}\). Since \(9\) is a perfect square, we further simplify it to \(\sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5} = 3\sqrt{5}\).

Step 3: Subtract the Simplified Terms

Now, with both radicals simplified, we have \(5\sqrt{5} - 3\sqrt{5}\). Since both terms have \(\sqrt{5}\) in common, we can combine them: \(5\sqrt{5} - 3\sqrt{5} = (5 - 3)\sqrt{5} = 2\sqrt{5}\).

Key Concepts

Radical Expressions

Radical expressions involve the use of roots, commonly square roots. Simplifying them means rewriting them in their simplest form. Consider the expression \( \sqrt{125} - \sqrt{45} \). We break each part into its prime factors or perfect squares to simplify:

• For \( \sqrt{125} \), recognize that 125 can be factored as \( 25 \times 5 \).
• Since 25 is a perfect square, we simplify \( \sqrt{25} \) to 5, resulting in \( 5\sqrt{5} \).
• Similarly, for \( \sqrt{45} \), use the factors \( 9 \times 5 \).
• The perfect square 9 simplifies to 3, giving us \( 3\sqrt{5} \).

Each step involves breaking down the number under the radical sign into its simplest parts, making it easier to perform operations on radical expressions later on.

Perfect Squares

A perfect square is an integer that is the square of another integer. Recognizing these can greatly help in simplifying radical expressions. In our exercise, we identified:

• 25 as a perfect square because \( 25 = 5^2 \).
• 9 as a perfect square because \( 9 = 3^2 \).

These perfect squares allow for simplification under the radical sign when simplifying expressions. By factoring numbers into elements where at least one is a perfect square, you can take its root out of the radical. This process dramatically simplifies complex expressions and is a crucial skill for ease and accuracy when working through algebraic problems.

Combining Like Terms

When dealing with radical expressions, combining like terms is similar to how you handle variables in algebra. After simplifying radicals, you may find terms with the same radical part - these are like terms. In our case:

• We have \( 5\sqrt{5} \) and \( 3\sqrt{5} \).
• Both terms have \( \sqrt{5} \), which can be thought of like a variable \( x \).
• Therefore, you can combine them: \( 5\sqrt{5} - 3\sqrt{5} = (5-3)\sqrt{5} = 2\sqrt{5} \).

The key is to ensure the radical part of each term is identical. Once confirmed, treat the radical as if it were a common variable and combine the coefficients. This technique helps in simplifying expressions to their most condensed form, allowing for more intuitive and streamlined computation.