Verified Answer :

  • Analyze the radicand 12 and identify its factors, noting that \(12 = 4 \times 3\).
  • Use the property \(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\) to simplify \(\sqrt{12}\) to \(2\sqrt{3}\).
  • Multiply the coefficient 5 by 2 to get the final simplified expression.
  • The final simplified form is \(\boxed{10\sqrt{3}}\).

Explanation

  1. Analyzing the Expression

The objective is to simplify the expression \(5\sqrt{12}\). To simplify a square root, we look for factors of the radicand (the number inside the root, which is 12) that are perfect squares. A perfect square is an integer that is the square of another integer, such as \(1, 4, 9, 16, \dots\). We want to find the largest perfect square that divides 12.

  1. Simplifying the Radical Part

We factor the number 12 by looking for its largest perfect square factor. Since \(12 = 4 \times 3\) and 4 is a perfect square (\(4 = 2^2\)), we can rewrite the radical:
\[\sqrt{12} = \sqrt{4 \times 3}.\]
Using the property of radicals that states \(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\) for non-negative \(a\) and \(b\), we get:
\[\sqrt{12} = \sqrt{4} \cdot \sqrt{3}.\]
Since \(\sqrt{4} = 2\), this simplifies to:
\[\sqrt{12} = 2\sqrt{3}.\]

  1. Final Simplification

Now, we substitute this back into the original expression:
\[5\sqrt{12} = 5 \cdot (2\sqrt{3}).\]
Multiplying the constant coefficients:
\[5 \cdot 2 = 10.\]
Therefore, the simplified form of the expression is \(10\sqrt{3}\).

Examples

Simplifying radicals is a foundational skill in geometry and physics. For example, when calculating the side length of a square given its area, or finding the distance between two points using the Pythagorean theorem, results often involve square roots. Simplifying these expressions to their most basic form, like \(10\sqrt{3}\) instead of \(5\sqrt{12}\), makes it much easier to compare values or perform further calculations in engineering and design projects.

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