Goal
The task “60 square root simplified” means rewriting \(\sqrt{60}\) so that no perfect-square factor remains inside the radical. The final answer should be in simplest radical form.
Key property
If \(a \ge 0\) and \(b \ge 0\), then \[ \sqrt{a\cdot b} = \sqrt{a}\cdot\sqrt{b}. \] In particular, if \(a\) is a perfect square, \(\sqrt{a}\) becomes an integer and can be moved outside the radical.
Method 1: Factor out the largest perfect square
Find a perfect-square factor of \(60\). Since \(60 = 4\cdot 15\) and \(4\) is a perfect square:
- Rewrite using a perfect square: \[ \sqrt{60} = \sqrt{4\cdot 15}. \]
- Split the radical: \[ \sqrt{4\cdot 15} = \sqrt{4}\cdot\sqrt{15}. \]
- Evaluate \(\sqrt{4}\): \[ \sqrt{4}\cdot\sqrt{15} = 2\cdot\sqrt{15}. \]
60 square root simplified: \(\;\sqrt{60} = 2\sqrt{15}\)
Method 2: Prime factorization and pairing
Prime factorization makes the perfect-square structure explicit:
\[ 60 = 2\cdot 2\cdot 3\cdot 5 = 2^2\cdot 3\cdot 5. \]
Use \(\sqrt{2^2\cdot 3\cdot 5} = \sqrt{2^2}\cdot\sqrt{3\cdot 5}\):
\[ \sqrt{60} = \sqrt{2^2\cdot 3\cdot 5} = 2\cdot\sqrt{15}. \]
Quick verification
Squaring the simplified expression should reproduce the original radicand:
\[ (2\sqrt{15})^2 = 2^2\cdot(\sqrt{15})^2 = 4\cdot 15 = 60. \]
Common pitfalls
- Stopping too early: \(\sqrt{60}=\sqrt{4\cdot 15}\) is not yet simplified until the perfect square \(4\) is extracted.
- Incorrect splitting: \(\sqrt{a+b}\ne \sqrt{a}+\sqrt{b}\) in general; only products split: \(\sqrt{a\cdot b}=\sqrt{a}\cdot\sqrt{b}\).
- Leaving a square inside: after simplification, the number inside the radical must have no perfect-square factor greater than \(1\).