Skill: Estimate square roots

To estimate a square root, find nearby perfect squares and use them as benchmarks. Remember that $\sqrt{n}$ means the principal, nonnegative number whose square is $n$.

Example: Estimate $\sqrt{50}$ to the nearest hundredth.

1. Find the perfect squares around $50$:

$$7^2=49 \qquad 8^2=64$$

Since $50$ is between $49$ and $64$,

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

2. Refine the estimate to tenths. Try $7.1$:

$$7.1^2=50.41$$

Because $50$ is between $7.0^2=49$ and $7.1^2=50.41$,

$$7.0<\sqrt{50}<7.1.$$

3. Refine to hundredths. Try $7.07$ and $7.08$:

$$7.07^2=49.9849$$ $$7.08^2=50.1264$$

Therefore,

$$49.9849<50<50.1264,$$

so

$$7.07<\sqrt{50}<7.08.$$

Thus, to the nearest hundredth,

$$\boxed{\sqrt{50}\approx 7.07}.$$

The estimate is not exactly equal to $\sqrt{50}$ because $50$ is not a perfect square.