√(64x²) is 8x, and the tempting wrong answers each take half of it out and leave the other half behind. These free printable worksheets show one of those half-finished lines and ask which answer is correct.
A root over a product has to be taken of every factor, and the four wrong answers are the four ways to stop early: the number rooted but not the x, the x rooted but not the number, neither, or the root dropped entirely. Each looks reasonable on its own and only looks wrong beside the others.
Why this sheet works
Some of the roots do not come out cleanly, and those are the useful ones. When a factor has no square in it the answer keeps a root — 5x√7 rather than 35x — and a student who has learned to expect a tidy answer will reach for the tidy wrong one. The answers here are deliberately not all the same shape.
How this one works
One question from a sheet like this one, worked a step at a time — with a sentence saying what just happened and why.
Read each line of working and choose the answer that is correct.
1Mia says √(4x²) + √(49x²) comes to x√53. Which is correct?x√5314x9x53xWhy?
2On Sam's paper: √(27x²) + √(3x²) = x√30. Which line should replace it?x√304x√64x√34xWhy?
3Sam wrote √(25x²) + √(36x²) = x√61. What should it be?11x30xx√6161xWhy?
4Check this line: √(96x²) + √(6x²) = 5x. Which is correct?5x5x√12x√1025x√6Why?
Algebra reasoning · Square rootsAnswer key
1. √(4x²) + √(49x²) = 9x, not x√53 (the two insides were added under one root before either was rooted)2. √(27x²) + √(3x²) = 4x√3, not x√30 (the two insides were added under one root before either was rooted)3. √(25x²) + √(36x²) = 11x, not x√61 (the two insides were added under one root before either was rooted)4. √(96x²) + √(6x²) = 5x√6, not 5x (the part that would not come out was dropped)
