I’m stuck in surd-land and my brain keeps trying to “smush” things that apparently don’t want to be smushed. I know there’s a rule about multiplying under a square root that seems to work nicely, so I keep thinking adding should behave the same. But when I try something like √2 + √8 and squish it into √10… my calculator gives me a very judgey side-eye.
Here’s what I did (which I’m pretty sure is wrong):
– I tried √2 + √8 = √10.
– I also tried to split √12 into √6 + √6, which felt clever for about three seconds.
– And while rationalising denominators, I even did 1/(1 + √3) = 1/(1 + 3) = 1/4, which… yeah, probably illegal.
I’m confused because multiplying seems to let me combine things inside the radical, so why doesn’t adding work the same way? What are the actual do’s and don’ts here? If I have a simple example like √2 + √8, what’s the right kind of move I should be thinking about (if any)? And is there a simple reason why the “add inside the root” trick fails that I can keep in my head?
Thanks for any nudges-I keep looping on this and would love a way to stop making the same oops.
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3 Responses
Short version: you can multiply or divide inside one square root, but you can’t add or subtract inside it. Why? If sqrt(a) + sqrt(b) were sqrt(a + b), then squaring both sides would give a + b + 2sqrt(ab) = a + b, which would force ab = 0-so it only “works” in trivial cases. A quick gut-check: sqrt(1) + sqrt(1) = 2, but sqrt(2) ≈ 1.414, so no. The right move with sqrt(2) + sqrt(8) is to simplify the radicals first: sqrt(8) = sqrt(4·2) = 2sqrt(2), so sqrt(2) + sqrt(8) = sqrt(2) + 2sqrt(2) = 3sqrt(2). For your denominator snafu, 1/(1 + sqrt(3)) isn’t 1/4; use the conjugate: multiply top and bottom by (1 − sqrt(3)) to get (1 − sqrt(3))/(1 − 3) = (sqrt(3) − 1)/2. Think of square roots as “side lengths of squares with a given area”: adding side lengths of two separate squares doesn’t magically give the side length of one bigger square with the combined area-that’s not how areas add. Mental rules to keep: do use sqrt(ab) = sqrt(a)sqrt(b) and sqrt(a/b) = sqrt(a)/sqrt(b) (b > 0), do pull out perfect squares and then combine like terms (k·sqrt(n) + m·sqrt(n) = (k + m)·sqrt(n)); don’t try sqrt(a + b) = sqrt(a) + sqrt(b) or split a single root across addition. A tidy refresher with examples: https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:radicals/x2f8bb11595b61c86:adding-and-subtracting-radicals/a/adding-and-subtracting-radicals-review.
Great instinct noticing there’s a “smush” rule for multiplication: sqrt(ab) = sqrt(a)·sqrt(b) (and similarly sqrt(a/b) = sqrt(a)/sqrt(b) for b>0). But addition is the trickster here: in general sqrt(a + b) ≠ sqrt(a) + sqrt(b). The clean reason is the cross term: if sqrt(a) + sqrt(b) were equal to sqrt(a + b), then squaring both sides would give a + b + 2·sqrt(ab) = a + b, which forces sqrt(ab) = 0, i.e., one of a or b is 0-only the trivial case works. That’s why your √2 + √8 ≠ √10. The right move is to simplify like radicals so you can combine them: sqrt(8) = sqrt(4·2) = 2·sqrt(2), so sqrt(2) + sqrt(8) = sqrt(2) + 2·sqrt(2) = 3·sqrt(2). Similarly, splitting a single radical into a sum (like trying to turn sqrt(12) into sqrt(6) + sqrt(6)) is a no-go; instead look for perfect-square factors to pull out. And for rationalizing denominators, use conjugates: 1/(1 + sqrt(3)) × (1 − sqrt(3))/(1 − sqrt(3)) = (1 − sqrt(3))/(1 − 3) = (sqrt(3) − 1)/2. The big picture: multiply/divide inside radicals is allowed; add/subtract is not, unless you’ve made the radicals “like” and can factor them. Hope this helps!
Addition can’t be “smushed” under a root because the square root isn’t linear-(√a + √b)^2 = a + b + 2√(ab), so √2 + √8 = √2 + 2√2 = 3√2 (not √10); when I first learned this, I kept trying to cram numbers under the radical until a lightbulb moment with simplifying √8 and, for denominators, using the conjugate trick: 1/(1 + √3) → (√3 − 1)/2. For a tidy rundown of the do’s (multiply/divide) and don’ts (add/subtract) with radicals, this helped me: https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:radicals/x2f8bb11595b61c86:radical-properties/a/properties-of-square-roots