Key takeaway: Simplifying radicals means rewriting a root so that the number under the radical sign has no perfect-square factor larger than 1, for example turning the square root of 180 into 6 times the square root of 5.
Grade 9 – Algebra 1N-RN.A.2Numbers and Operations
Definition of Simplifying Radicals
Simplifying radicals rewrites a square root in its neatest exact form. A root such as \(\sqrt{180}\) is not wrong, but it hides a perfect square inside: \(180 = 36 \times 5\). Pulling the square root of 36 outside the radical leaves \(6\sqrt{5}\).
A radical is fully simplified when the number under the sign has no factor that is a perfect square (4, 9, 16, 25, 36 and so on) other than 1, when no fraction sits under the sign, and when no radical stays in a denominator. Simplified forms let you compare roots, add like radicals and give exact answers instead of rounded decimals.
Formula and notation
Simplifying radicals relies on the product rule for square roots, valid for \(a \ge 0\) and \(b \ge 0\):
\[\sqrt{ab} = \sqrt{a}\cdot\sqrt{b}\]
Choose \(a\) as the largest perfect square dividing the number. The result has the form \(k\sqrt{m}\), with \(k\) the whole-number coefficient outside and \(m\) the remaining radicand.
Key facts
- When simplifying radicals, look for the largest perfect-square factor: 4, 9, 16, 25, 36, 49, 64 or 81.
- Factor trees show pairs; each pair of equal factors moves outside as one copy.
- The product rule works for multiplication under the root, not for addition.
- Check by squaring: the coefficient squared times the radicand equals the original number.
Simplifying Radicals example
Simplify \(\sqrt{180}\) and give a decimal check to three places.
-
Find the largest perfect square dividing 180: \(36\), since \(180 = 36 \times 5\).
-
Split the root: \(\sqrt{180} = \sqrt{36}\cdot\sqrt{5} = 6\sqrt{5}\).
-
Check: \(\sqrt{180} \approx 13.416\) and \(6 \times 2.236 \approx 13.416\).
\(\sqrt{180} = \) \(6\sqrt{5}\), about 13.416.
Common mistake
Watch out
Wes spots only the factor 9 and writes \(\sqrt{180} = \sqrt{9 \cdot 20} = 3\sqrt{20}\), then stops.
Why it is wrong
The answer is correct but not finished: 20 still contains the perfect square 4. Continue with \(3\sqrt{20} = 3\sqrt{4 \cdot 5} = 3 \cdot 2\sqrt{5} = 6\sqrt{5}\). Starting with the largest perfect-square factor, 36, avoids the extra step.
Related terms and lessons
- Square Root
- Perfect Square
- Rational Exponents
- Prime Factorization
- Irrational Number
- Pythagorean Theorem
Browse the glossary: A–Z list · Algebra 1 vocabulary · Numbers and Operations