Verified Answer :

  • Find the prime factors of 45: \(45 = 9 \times 5 = 3^2 \times 5\).
  • Apply the property of radicals \(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\).
  • Simplify the radical: \(\sqrt{9 \times 5} = \sqrt{9} \cdot \sqrt{5} = 3\sqrt{5}\).
  • The simplified form is \(\boxed{3\sqrt{5}}\).

Explanation

  1. Understanding the Problem

The expression to simplify is \(\sqrt{45}\). To simplify a square root, we look for perfect square factors within the number under the radical, known as the radicand. A perfect square is an integer that is the square of another integer, such as \(1, 4, 9, 16, 25, \dots\). We aim to write the radicand as a product of a perfect square and another factor.

  1. Factoring the Radicand

We perform the prime factorization of 45. We know that \(45 = 9 \times 5\). Since 9 is a perfect square (\(3^2 = 9\)), we can rewrite the expression as \(\sqrt{9 \times 5}\).

  1. Simplifying the Expression

Using the radical property \(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\) for \(a, b \geq 0\), we split the expression:
\[\sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5}\]
Since \(\sqrt{9} = 3\), the expression simplifies to \(3\sqrt{5}\). This is the simplest radical form because 5 has no square factors other than 1.

Examples

Simplifying radicals is a foundational skill in geometry and physics. For example, if you need to calculate the hypotenuse of a right-angled triangle with sides \(3\) and \(6\), the Pythagorean theorem gives \(\sqrt{3^2 + 6^2} = \sqrt{9 + 36} = \sqrt{45}\). By simplifying this to \(3\sqrt{5}\), you can easily estimate the decimal value (since \(\sqrt{5} \approx 2.236\), the result is \(\approx 6.708\)) and work with it more cleanly in further calculations.

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