Estimate square roots

Square-root estimation involves locating the principal nonnegative value whose square equals or approximates a given nonnegative number, using nearby perfect squares as benchmarks. A learner can bound roots of integers and familiar decimals between consecutive whole numbers and refine those bounds to tenths or hundredths by comparing squared estimates, while understanding that non-perfect-square roots are generally irrational and that the square-root operation reverses squaring only for nonnegative values.

Detailed Explanation: 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.

Find the perfect squares around $50$ :

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

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

7<\sqrt{50}<8.

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.

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,

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.

Learn by doing: Estimate square roots

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Square Roots Approximating

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