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

The value of `\sqrt{`

lies between which two consecutive integers?`Q`}

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

`N`

`N` `< \sqrt{`

`Q`} <`N + 1`

`6`

Consider the perfect squares near

.
[What are perfect squares?]
`Q`

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`

is the nearest perfect square less than `N * N`

.`Q`

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

.`Q`

So, we know

.`N * N` < `Q` < `(N + 1) * (N + 1)`

So, `\sqrt{`

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

So `\sqrt{`

is between `Q`}

and `N`

.`N + 1`