√(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 this sheet, worked through a step at a time.
Mia says √(4x²) + √(49x²) comes to x√53. Which is correct?
√(4x²) + √(49x²)
√(4x²) + √(49x²)=2x + 7x
√(4x²) + √(49x²)=2x + 7x
√(4x²) + √(49x²)=9x
√(4x²) + √(49x²)=9x
they wrote:x√53
Do not check by reading their line. Somebody else's working is the most persuasive thing on the page, and rereading it is how you end up agreeing with it — do the step yourself first, then compare.
On your own, and writing the step nobody writes down: √(4x²) + √(49x²) becomes 2x + 7x.
Tidy that up and it is 9x.
Now compare. They wrote x√53, and the two insides were added under one root before either was rooted — so the correct answer is 9x.
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)
