Verified Answer :

  • Recognize that the radicand \(125\) can be expressed as the product of a perfect square and another number: \(125 = 25 \times 5\).
  • Apply the radical property \(\sqrt{ab} = \sqrt{a}\sqrt{b}\) to rewrite the expression as \(2 \times \sqrt{25} \times \sqrt{5}\).
  • Evaluate the square root of the perfect square: \(\sqrt{25} = 5\).
  • Multiply the constants to arrive at the final simplified value of \(\boxed{10\sqrt{5}}\).

Explanation

  1. Analyzing the Expression

We are given the expression \(2\sqrt{125}\) and need to simplify it. To simplify a square root, we look for perfect square factors within the radicand (the number inside the square root symbol). We know that the square root of a product is equal to the product of the square roots, meaning \(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\) for \(a, b \geq 0\). This property allows us to extract perfect squares from the radical.

  1. Factoring the Radicand

We factor 125 to find its perfect square components. Since \(125 = 25 \times 5\), and 25 is a perfect square (\(5^2 = 25\)), we can rewrite the expression as:
\[2\sqrt{125} = 2\sqrt{25 \times 5}\]
Applying the property of radicals mentioned earlier, this becomes:
\[2\sqrt{25} \times \sqrt{5}\]

  1. Final Simplification

Now, we evaluate the square root of 25:
\[\sqrt{25} = 5\]
Substituting this back into our expression, we get:
\[2 \times 5 \times \sqrt{5}\]
Multiplying the constant terms, we find:
\[10 \times \sqrt{5} = 10\sqrt{5}\]
Thus, the simplified form of the original expression is \(10\sqrt{5}\).

Examples

Imagine you are designing a garden and you need to calculate the length of a support beam for a trellis. If the geometry of the trellis requires a beam of length \(\sqrt{125}\) meters, it is much easier to work with the value \(5\sqrt{5}\) meters (which is approximately \(5 \times 2.236 \approx 11.18\) meters) to estimate how much material you need. Simplifying radicals helps bridge the gap between abstract mathematical expressions and practical measurements, allowing you to quickly visualize or estimate values in construction, engineering, or everyday DIY projects.

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