Simplifying Radicals
In short: Simplifying radicals means rewriting a square root so that no perfect square factor remains under the radical sign, for example √72 becomes 6√2.
Algebra 1Numbers and OperationsN-RN.A.2
What is Simplifying Radicals?
Although \(\sqrt{72}\) is exact, it looks untidy, and simplifying radicals pulls out every perfect square hiding inside it, leaving a whole number in front and a smaller number under the sign.
The tool is the product rule, \(\sqrt{ab} = \sqrt{a}\cdot\sqrt{b}\) for \(a, b \ge 0\). Since \(72 = 36 \cdot 2\) and \(\sqrt{36} = 6\), you get \(\sqrt{72} = 6\sqrt{2}\). A radical is simplified when its radicand has no square factor greater than 1.
How to spot it
- Simplifying radicals is needed when an answer is left with a big radicand such as 72 or 180.
- The instructions say simplify, write in simplest radical form or give an exact answer.
- A Pythagorean theorem result is not a perfect square, so the side length stays a radical.
- Knowing your squares up to 144 helps you test the radicand quickly.
The method
- Run through the perfect squares you know and find the largest one that divides the radicand evenly.
- Split the radicand into that square times the leftover factor.
- Take the square root of the perfect square and write it outside the radical.
- Check that the number left inside has no perfect square factor; if it does, repeat.
Simplifying Radicals: solved example
Problem
A square garden plot has an area of 180 square feet. State the side exactly, with a radical, and then approximate it with two decimal places.
Solution
The side is \(\sqrt{180}\). The largest perfect square dividing 180 is 36, since \(180 = 36 \cdot 5\).
So \(\sqrt{180} = \sqrt{36}\cdot\sqrt{5} = 6\sqrt{5} \approx 6(2.2361) \approx 13.42\). The side is \(6\sqrt{5}\) feet, about 13.42 feet.
Try it
Question 1
Simplify \(\sqrt{48}\).
Show the answer
\(48 = 16 \cdot 3\), so \(\sqrt{48} = 4\sqrt{3}\).
Question 2
Simplify \(3\sqrt{50}\).
Show the answer
\(\sqrt{50} = \sqrt{25}\cdot\sqrt{2} = 5\sqrt{2}\), so \(3\sqrt{50} = 15\sqrt{2}\).
Simplifying Radicals vs. Square Root
A plain square root asks for the number that multiplies by itself to give the radicand. Simplification does not change the value; it only rewrites an irrational root in a cleaner form, such as 6√2 for √72, so answers can be compared and added with like radicals.
Keep going
Lessons
Worksheets and quizzes
- Exponent Rules and Radicals – Worksheet
- Exponent Rules and Radicals – Quiz
- Square Roots and Cube Roots – Worksheet
- Square Roots and Cube Roots – Quiz
Related words
- Square Root
- Perfect Square
- Irrational Number
- Rational Exponents
- Pythagorean Theorem
- Prime Factorization
Whole dictionary: A–Z · Algebra 1 · Numbers and Operations