Estimate √20 to the nearest tenth
- 4² = 16 and 5² = 25, so 4 < √20 < 5.
- 4.4² = 19.36 and 4.5² = 20.25.
- The midpoint 4.45 has square 19.8025, below 20. Therefore √20 > 4.45 and rounds to 4.5.
Estimate square roots
Bound an unknown root, then test the precision.
Bracket the root with known perfect squares
Bracket a root with perfect squares and justify a decimal estimate by squaring.
Since 49 < 60 < 64, taking principal square roots gives 7 < √60 < 8. These benchmarks provide a reasonableness check before a calculator result. A square of area 60 units² has a side between 7 and 8 units, even though its side is not a whole number.
Explore the relationship
Predict first. Change one value, then explain the result. Diagrams show the entered dimensions; displayed decimals are rounded.
Refine an estimate by squaring trial values
For √60, 7.7² = 59.29 and 7.8² = 60.84, so the root lies between 7.7 and 7.8. To round to the nearest tenth, test the midpoint 7.75: 7.75² = 60.0625. Since 60 is smaller, √60 < 7.75 and rounds to 7.7. Simply knowing the two adjacent tenths is not always enough to choose the nearer one.
Distinguish a bound, estimate and exact expression
√60 is an exact expression. The decimal 7.746 is an approximation. Write √60 ≈ 7.7 to the nearest tenth, not an equality. Do not assume the root lies the same fraction of the way between integers as the area does between square numbers: squaring does not grow linearly. Round only the final answer in a multi-step calculation.
Worked examples
Bound √90 and check a decimal estimate
- 9² = 81 < 90 < 100 = 10².
- 9.4² = 88.36 and 9.5² = 90.25.
- 9.45² = 89.3025 < 90, so √90 rounds to 9.5 to the nearest tenth.
Guided Practice
Estimate √50 to the nearest tenth. Justify the rounding.
A first hint
First locate it between 7 and 8, then compare with the midpoint 7.05.
Compare your reasoning
7.05² = 49.7025 < 50 and 7.1² = 50.41 > 50. The root is between 7.05 and 7.1, so it rounds to 7.1.
Practice
Fresh questions · useful feedback
Build fluency, one decision at a time.
Work through eight questions. Two independent correct answers introduce a harder version; supported answers bring the next question back to the foundation. You can use hints at any time.
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Connections
A student says √30 is exactly halfway between 5 and 6 because 30 is near the middle of 25 and 36. Test 5.5² and explain the error.
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