The expression square root of 24 refers to the positive number whose square equals \(24\). Since \(24\) is not a perfect square, \(\sqrt{24}\) is irrational, but it can be simplified by factoring out perfect-square parts.
Goal Write \(\sqrt{24}\) in simplest radical form, meaning no perfect square factor remains inside the radical.
Simplifying \(\sqrt{24}\) using perfect-square factors
Factor \(24\) to expose a perfect square. A convenient factorization is \(24=4\cdot 6\), where \(4\) is a perfect square.
\[ \sqrt{24}=\sqrt{4\cdot 6}. \]
Use the product rule for square roots: \(\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}\) for \(a\ge 0\) and \(b\ge 0\).
\[ \sqrt{24}=\sqrt{4}\cdot\sqrt{6}=2\sqrt{6}. \]
Simplest form \(\sqrt{24}=2\sqrt{6}\).
Verification by squaring
A quick algebra check confirms the simplification is correct:
\[ \left(2\sqrt{6}\right)^2 = 2^2\cdot\left(\sqrt{6}\right)^2 = 4\cdot 6 = 24. \]
Decimal approximation of \(\sqrt{24}\)
Since \(\sqrt{24}=2\sqrt{6}\) and \(\sqrt{6}\approx 2.44949\), the approximation is:
\[ \sqrt{24}=2\sqrt{6}\approx 2\cdot 2.44949=4.89898\approx 4.899. \]
Visualization: locating \(\sqrt{24}\) on a number line
Because \(4^2=16\) and \(5^2=25\), \(\sqrt{24}\) lies between \(4\) and \(5\), very close to \(5\). The mark below places \(\sqrt{24}\approx 4.899\) on a number line segment from \(4\) to \(5\).
Summary
The square root of 24 simplifies by factoring out the perfect square \(4\): \(\sqrt{24}=\sqrt{4\cdot 6}=2\sqrt{6}\), and numerically \(\sqrt{24}\approx 4.899\).