What is the simplified exact value of √18 - √2?

Introduction / Context:
This question asks for the simplification of an expression involving square roots. Problems of this type appear very often in aptitude tests and algebra practice because they check understanding of how to rewrite surds in simplified radical form without decimals.

Given Data / Assumptions:

• We must simplify the expression √18 - √2.
• The goal is to express the result in simplest radical form.
• No decimal approximations are required; we keep exact surds.

Concept / Approach:
The standard technique for simplifying square roots is to factor the number under the root into a perfect square times another factor. We then take the square root of the perfect square outside the radical. For example, 18 can be written as 9 * 2, where 9 is a perfect square. We also look for common factors and try to express both surds in terms of the same basic root to combine like terms, much like combining like algebraic terms.

Step-by-Step Solution:
Step 1: Rewrite √18 using a perfect square factor: 18 = 9 * 2. Step 2: Therefore, √18 = √(9 * 2) = √9 * √2 = 3√2. Step 3: The expression now becomes 3√2 - √2. Step 4: Factor out √2: 3√2 - √2 = (3 - 1)√2. Step 5: Simplify the coefficient: (3 - 1)√2 = 2√2. Step 6: Thus, the simplified exact value is 2√2.

Verification / Alternative check:
As a check, we may approximate the values. √18 is about 4.243 and √2 is about 1.414, so √18 - √2 is approximately 2.829. The expression 2√2 is about 2 * 1.414 = 2.828, which is very close, with any difference due only to rounding. This confirms that the simplification is correct.

Why Other Options Are Wrong:

• Option B: 5/√2 is approximately 3.536, which is larger than the true value.
• Option C: 5√2 is about 7.071, far too large.
• Option D: 2√5 is about 4.472, also much larger.
• Option E: 2/√5 is about 0.894, which is much smaller than the correct value.

Common Pitfalls:
Students sometimes try to compute decimal values too early or forget to simplify 18 correctly as 9 * 2. Another error is to think that √18 - √2 equals √(18 - 2), which is not allowed because square root is not a linear operation. Always simplify each radical separately and combine only like surds.

Final Answer:
The simplified value of the expression is 2√2.