How do I simplify a radical when the number has a perfect-square factor?
Quick answer
Find a perfect-square factor of the number under the square root, rewrite the number as that factor times what remains, and take the square root of the perfect-square factor. For example, 72 = 36 × 2, so √72 = 6√2. Check by squaring: (6√2)² = 36 × 2 = 72. This method applies to nonnegative radicands.[S1]
The question, the details, the next step.
What should I look for when a number under a square root is not itself a perfect square?
Look for a perfect-square factor—a factor such as 4, 9, 16, 25, or 36 whose square root is a whole number. Split the radicand (the number under the radical sign) into that factor and the leftover factor. Then take the square root of the perfect-square part and leave the other part under the radical. For nonnegative factors, this uses the product property of square roots.[S1]
For example, 12 = 4 × 3. So √12 = √4 × √3 = 2√3. The 3 stays under the radical because it has no perfect-square factor greater than 1.
How do I choose the factor if there are several possibilities?
Choose the largest perfect-square factor that divides the radicand evenly. That usually gets you to a simplified result in one step. If you choose a smaller one, you can simplify further afterward.
Take √72. The number 72 is divisible by 36, which is a perfect square:
- Rewrite: 72 = 36 × 2.
- Split the radical: √72 = √36 × √2.
- Take the square root of 36: √72 = 6√2.
A smaller choice also works: 72 = 4 × 18 gives √72 = 2√18, and then √18 = 3√2, so the result is again 6√2. The larger square factor just avoids an extra round.
Can you show another example and how to check the simplified answer?
For √45, use 45 = 9 × 5. Then √45 = √9 × √5 = 3√5.
To check, square the proposed answer: (3√5)² = 3² × (√5)² = 9 × 5 = 45. It squares back to the original radicand, so the result is consistent with √45. This check uses the square-root product rule for nonnegative factors.[S1]
The same approach works for √12: (2√3)² = 4 × 3 = 12. Remember that the simplified answer can still contain a radical; the goal is to remove any perfect-square factor from under it.
What if I cannot see a perfect-square factor right away? Is there a routine I can follow?
Try the integer squares from largest to smaller, stopping when you find one that divides the radicand evenly. For a smaller radicand, skip any square larger than it. For example, with √80, 16 divides 80: 80 = 16 × 5, so √80 = √16 × √5 = 4√5. Check: (4√5)² = 16 × 5 = 80.
Copyable setup:
√n = √(p × r) = √p × √r = k√r
Choose p = k² so that p × r = n. If the leftover factor r still has a perfect-square factor greater than 1, simplify again. The rule here is for square roots of nonnegative real numbers; it does not establish how to simplify other kinds of radicals.[S1]
A practical checklist
Write down the original radicand.
Find a perfect-square factor that divides it evenly.
Rewrite it as the perfect square times the remaining factor.
Take the square root of the perfect-square factor and keep the remainder under the radical.
Square your result and confirm it gives the original radicand.
A few more useful answers
Does the leftover factor have to be prime?
No. It only needs to have no perfect-square factor greater than 1 left to take out. If it does have one, simplify again.
Can I split every square root across multiplication?
For nonnegative factors, the product property lets you split a square root of a product into the product of the square roots.[S1] The cited rule does not establish a general rule for other kinds of radicals.
What does it mean if the radicand has no perfect-square factor greater than 1?
There is no whole-number square-root factor to take out, so the radical is already simplified by this method. You can still check a proposed simplification by squaring it.
Sources
References used for this guide.
- [S1] OpenStax Elementary Algebra 2e: Chapter 9 Key Concepts on Square Roots openstax.org · Checked