Estimate square roots between consecutive integers

A learner understands that the principal square root of a nonnegative integer lies between consecutive integers whose squares bound the number: if $n^2 < m < (n+1)^2$ , then $n<\sqrt m<n+1$ . This distinguishes perfect-square roots from irrational roots and supports locating roots on a number line; this scope does not include methods for high-precision decimal approximations or roots of more general expressions.

Detailed Explanation: Estimate square roots between consecutive integers

To estimate $\sqrt{m}$ between consecutive integers:

Find a perfect square less than $m$ .
Find the next perfect square greater than $m$ .
Use their square roots as the bounds.

Example: Estimate $\sqrt{50}$

Find the consecutive perfect squares around $50$ :

7^2=49

and

8^2=64

Since

49<50<64,

we can take the square root of each part:

\sqrt{49}<\sqrt{50}<\sqrt{64}.

Therefore,

\boxed{7<\sqrt{50}<8}.

So, $\sqrt{50}$ is between $7$ and $8$ . It is not a whole number because $50$ is not a perfect square. On a number line, place $\sqrt{50}$ somewhere between $7$ and $8$ .

Learn by doing: Estimate square roots between consecutive integers

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Number Types (Irrational) - Between X and Y - Positive Square Roots

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