Nested Radicals: Step-by-Step Guide
A nested radical has a square root inside another square root, like the square root of (a + b root c). Sometimes it can be rewritten as a simpler sum, root x + root y. The trick is to find two numbers x and y that add up to a and multiply to (b^2 x c)/4.
Math Racer skill guide · Algebra 2 · Grade 11
How to solve it, step by step
- Guess the form. Assume that the square root of (a + b root c) = root x + root y, for two numbers x and y that you still need to find.\[\sqrt{a + b\sqrt{c}} = \sqrt{x} + \sqrt{y}\]
- Square both sides and match the parts. Squaring gives a + b root c = x + y + 2 root (xy). Match the plain-number parts: x + y = a. Match the root parts: 2 root (xy) = b root c, so 4xy = b^2 x c.\[x+y = a,\quad 4xy = b^{2}c\]
- Find x and y. You need two numbers that add up to a and multiply to (b^2 x c)/4. This is the same kind of puzzle as factoring a trinomial. It only works nicely when a^2 - b^2 c is a perfect square.\[x,y = \frac{a \pm \sqrt{a^{2}-b^{2}c}}{2}\]
- Write the answer and check. Write root x + root y. Square it to make sure you get the original expression back.\[\sqrt{a + b\sqrt{c}} = \sqrt{x} + \sqrt{y}\]
Worked example
Denest: sqrt(7 + 4 sqrt(3))
Step 1: Assume the square root of (7 + 4 root 3) = root x + root y.
Step 2: Square both sides: 7 + 4 root 3 = x + y + 2 root (xy). Match the parts: x + y = 7, and 2 root (xy) = 4 root 3, so root (xy) = 2 root 3, which means xy = 12.
Step 3: Find two numbers that add to 7 and multiply to 12. Those are 4 and 3.
Step 4: So the square root of (7 + 4 root 3) = root 4 + root 3 = 2 + root 3.
Check: (2 + root 3)^2 = 4 + 4 root 3 + 3 = 7 + 4 root 3. It matches.
Tips and common mistakes
- This only works nicely when a^2 - b^2 c is a perfect square. If not, the expression cannot be simplified this way.
- When you square root x + root y, the middle term 2 root (xy) is where the inner root comes from.
- Write the pair of equations x + y = a and xy = (b^2 c)/4. This turns the problem into a factoring puzzle.
- Always check by squaring your answer.
- If the inside has a minus sign, try root x - root y, where root x is the bigger one.
- Often it helps to write the middle term as 2 times a root first, like 7 + 2 root 12. Then you look for two numbers that add to 7 and multiply to 12.
- A common pattern: a + 2 root b can be simplified when you can split a into two numbers that multiply to b.