Verified Answer :

  • Identify that the radicand 147 can be factored into \(49 \times 3\), where 49 is a perfect square.
  • Apply the radical rule \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\) to simplify \(\sqrt{147}\) to \(7\sqrt{3}\).
  • Multiply the coefficient 3 by the resulting 7 to get the final simplified expression \(\boxed{21\sqrt{3}}\).

Explanation

  1. Analyzing and Factoring the Radicand

To simplify the expression \(3 \sqrt{147}\), we first analyze the radicand, which is 147. We look for factors of 147 that are perfect squares. A common method is to perform prime factorization: 147 is divisible by 3 since the sum of its digits, \(1 + 4 + 7 = 12\), is divisible by 3. Dividing 147 by 3 gives 49, so \(147 = 3 \times 49\). Recognizing that \(49 = 7^2\), we see that 49 is a perfect square.

  1. Simplifying the Square Root

Now we use the property of radicals which states that for any non-negative numbers \(a\) and \(b\), \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\). We can rewrite the radical part as:
\[\sqrt{147} = \sqrt{49 \times 3} = \sqrt{49} \times \sqrt{3} = 7 \sqrt{3}\]
Substituting this result back into our original expression, we get:
\[3 \sqrt{147} = 3 \times (7 \sqrt{3})\]

  1. Final Calculation and Simplification

Finally, we perform the multiplication to complete the simplification:
\[3 \times 7 \sqrt{3} = 21 \sqrt{3}\]
Thus, the simplified form of the expression is \(21\sqrt{3}\).

Examples

Simplifying square roots is very useful in geometry when working with lengths and areas. For example, if you are calculating the side length of a triangle using the Pythagorean theorem and arrive at \(\sqrt{147}\), simplifying it to \(7\sqrt{3}\) makes it much easier to estimate its value (using \(\sqrt{3} \approx 1.732\)) or to combine it with other similar terms in more complex algebraic expressions. This skill is foundational in trigonometry and higher-level mathematics.

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