Verified Answer :

  • Factor the number inside the square root to find perfect square factors: \(20 = 4 \times 5\).
  • Simplify the radical by taking the square root of the perfect square: \(\sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5}\).
  • Multiply by the original coefficient: \(7 \times 2\sqrt{5} = 14\sqrt{5}\).
  • The final simplified expression is \(\boxed{14\sqrt{5}}\).

Explanation

  1. Factorizing the Radicand

To simplify the expression \(7\sqrt{20}\), we need to extract the largest perfect square factor from the number under the square root. The number 20 can be factored into prime numbers: \(20 = 2 \times 2 \times 5 = 2^2 \times 5\). Since \(2^2 = 4\) is a perfect square, we can use the property \(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\) to simplify the expression.

  1. Simplifying the Radical and Substituting

Using the property of square roots, we can rewrite the radical \(\sqrt{20}\) as:
\[\sqrt{20} = \sqrt{4 \times 5} = \sqrt{4} \cdot \sqrt{5} = 2\sqrt{5}.\]
Now, substitute this back into the original expression:
\[7\sqrt{20} = 7 \times (2\sqrt{5}).\]

  1. Final Simplification

Finally, multiply the coefficients:
\[7 \times 2\sqrt{5} = 14\sqrt{5}.\]
Thus, the simplified form of the expression is \(14\sqrt{5}\).

Examples

Simplifying radicals is extremely useful in fields like architecture and engineering, where you often need to calculate dimensions or distances using the Pythagorean theorem. For example, if you are designing a support beam for a roof and the length involves a calculation like \(\sqrt{20}\) meters, converting it to \(2\sqrt{5}\) meters (approximately \(4.47\) meters) makes it much easier to handle the math when adding or comparing lengths during construction. Keeping it in radical form is also more precise than using a decimal approximation.

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