Estimating square roots as between two roots Common Core State Standard: approximately 8.NS Video coverage: http://www.khanacademy.org/test-prep/cahsee/v/cahsee-practice--problems-1-3 question 2 Prerequisite: Square roots
randRange( 2, 11 ) randRange( N * N + 1, (N + 1) * (N + 1) - 1 )

The value of \sqrt{Q} lies between which two consecutive integers?

Integers that appear in order when counting, for example 2 and 3.

N

N < \sqrt{Q} < N + 1

two integers, like 6

Consider the perfect squares near Q. [What are perfect squares?]

Perfect squares are integers which can be obtained by squaring an integer.

The first 13 perfect squares are:

\qquad 1,4,9,16,25,36,49,64,81,100,121,144,169

N * N is the nearest perfect square less than Q.

(N + 1) * (N + 1) is the nearest perfect square more than Q.

So, we know N * N < Q < (N + 1) * (N + 1).

So, \sqrt{N * N} < \sqrt{Q} < \sqrt{(N + 1)*(N + 1)}.

So \sqrt{Q} is between N and N + 1.