Simplifying Square Roots

Example:

Find the square root.

$\sqrt{49}$

$\sqrt{49}$ indicates the positive square root of $49$ .

Since $7\times 7$ is $49$ , $\sqrt{49}=7$ .

Find the square root.

$-\sqrt{25}$

$-\sqrt{25}$ indicates the negative square root of $25$ .

Since $5\times 5$ is $25$ , $-\sqrt{25}=-5$ .

Example:

Estimate $\sqrt{69}$ to the nearest whole number.

First list some perfect squares and look for the whole numbers near $69$ .

$\text{1, 4, 9, 16, 25, 36, 49, 64, 81, 100, }\mathrm{...}$

Observe that $\text{69}$ lies between the perfect squares $\text{64}$ and $\text{81}$ . That is, $64<69<81$ .

Now find the square root of each number.

$\begin{array}{l}\sqrt{64}<\sqrt{69}<\sqrt{81}\\ 8<\sqrt{69}<9\end{array}$

So, $\sqrt{69}$ is between $8$ and $9$ .

Since $69$ is much closer to $64$ than to $81$ , the best whole number estimate is $8$ .