The radical sign is like a prison. Twelve can be expressed as a product of prime factors so √12 = √(2×2×3). The 2’s pair up and try to break out. Sadly, only one of them survives the escape. √12 becomes 2√3.
That’s how I was taught to simplify radicals. No joke.
I imagined the numbers yelling “All in the name of liberty! Got to be free! JAlLBREAK!” as they scaled the prison walls. To this day, I can’t get this song out of my head when I teach this topic.
Many students are shown this method, albeit without the prison imagery. Write the prime factorization of the number. Circle the pairs. Write/multiply circled numbers outside the radical sign. There is real math behind this procedure. By definition, √2 × √2 = 2. However, I found that students who were taught this method couldn’t tell me why √(2×2×3) = 2√3. Where did the other 2 go?
Instead, I asked students to evaluate √12, then 2√3, using their calculators. Why are they equivalent? Students factored √12 as √4 × √3 (with some scaffolding for some). They understood where the 2 came from. Some began by factoring √12 as √6 × √2. Correct, but not helpful. The importance of finding factors that are perfect squares was discussed.
Marc Garneau shared with me his visual approach to simplifying radicals.
Consider a square with an area of 24. The side has length √24.
This square can be divided into 4 smaller squares, each with an area of 6. The sides of these smaller squares have length √6. Two of these lengths make up the side length of the large square, so √24 = 2√6.
24 can also be divided into 3 rectangles, each with an area of 8. Again, correct, but not helpful. How to simplify √45 as 3√5 and √72 as 6√2 are also shown above. Again, factors that are perfect squares are key.
I think it would be interesting to try this out. Some students may prefer this method, but most students will likely move towards simplifying radicals without drawing pictures. But by drawing pictures as they are learning this skill, students will be connecting mathematical ideas and building conceptual understanding. New learning (simplifying radicals in Math 10) will be connected to prior learning (concept of a square root introduced in Math 8). Students will have a more solid understanding of why perfect squares are used.