Simplify.
$\sqrt{125}$
$\sqrt{125}$
Simplify.
$\sqrt{125}$
To simplify the radical expression \(\sqrt{125}\), we first analyze the radicand, which is the number 125. Our objective is to write this radical in its simplest form, which involves extracting any perfect square factors from the radicand. A perfect square factor is a number that is the square of an integer (such as 1, 4, 9, 16, 25, ...). By identifying the largest perfect square factor of 125, we can simplify the expression using the property of radicals \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\) for \(a, b \geq 0\).
We perform the prime factorization of 125. We know that:
\[125 = 5 \times 5 \times 5 = 5^2 \times 5 = 25 \times 5.\]
By identifying 25 as a perfect square factor, we can rewrite the expression \(\sqrt{125}\) as:
\[\sqrt{125} = \sqrt{25 \times 5}.\]
Applying the radical property, we split the radical into two parts:
\[\sqrt{25 \times 5} = \sqrt{25} \times \sqrt{5}.\]
Since \(\sqrt{25} = 5\), the expression simplifies to:
\[5\sqrt{5}.\]
Thus, the simplest form of \(\sqrt{125}\) is \(5\sqrt{5}\).
Simplifying radicals is a fundamental skill in geometry and physics. For example, if you are calculating the length of the diagonal of a square with a side length of \(5\) units, the diagonal is given by the formula \(s\sqrt{2}\). If you have a square with an area of \(125\) square units, its side length is \(\sqrt{125}\), which simplifies to \(5\sqrt{5} \approx 11.18\) units. Mastering radical simplification allows for more precise measurements and easier manipulation of formulas in architecture, engineering, and various scientific fields.