Algebra
Exponents and scientific notation
Radical expressions
20 practice questions
2 video lessons
Theory + worked examples
Theory
A radical expression contains a root. Simplify by pulling out perfect squares:
\[\sqrt{ab}=\sqrt a\,\sqrt b.\]
Factor out the largest perfect square to simplify a square root.
Factor out a perfect square.
Radical rules.
The product rule:
\[\sqrt{ab}=\sqrt a\,\sqrt b,\qquad (\sqrt a)^2=a\]
Look for perfect-square factors like 4, 9, 16, 25, 36.
How to simplify a radical
- Find the largest perfect-square factor.
- Split the radical with the product rule.
- Take the root of the perfect square.
- Leave the rest under the radical.
Example 1 — Simplify
Simplify \(\sqrt{50}\).
Solution
Factor out the largest perfect square.
| \(\sqrt{50}\) | \(=\) | \(\sqrt{25}\cdot\sqrt2\) |
| \(=\) | \(5\sqrt2\) |
Example 2 — Multiply radicals
Simplify \(\sqrt8\cdot\sqrt2\).
Solution
Combine under one root.
| \(\sqrt{16}\) | \(=\) | \(4\) |
Example 3 — With a variable
Simplify \(\sqrt{x^2}\) (with \(x\ge0\)).
Solution
The square root undoes the square.
| \(\sqrt{x^2}\) | \(=\) | \(x\) |
Example 4 — A larger radical
Simplify \(\sqrt{72}\).
Solution
\(72=36\cdot2\).
| \(\sqrt{72}\) | \(=\) | \(6\sqrt2\) |
Common pitfalls
Factor out perfect squares, not just any factor.
\(\sqrt a+\sqrt b\neq\sqrt{a+b}\).
Simplify fully — no perfect-square factor should remain.
Frequently asked questions
How do you simplify \(\sqrt{50}\)?
Factor out \(25\): \(5\sqrt2\).
Is \(\sqrt{a}\,\sqrt{b}=\sqrt{ab}\)?
Yes for non-negative \(a,b\).
Does \(\sqrt{a+b}=\sqrt a+\sqrt b\)?
No — that is a common mistake.
Simplify \(\sqrt{18}\).
\(3\sqrt2\).
More in Exponents and scientific notation