A square has area 225 m²
- Side length = √225.
- 15 × 15 = 225, so the side is 15 m.
- Check: 15² m² = 225 m².
Square roots & inverse thinking
Recover a side length from a square area.
The square root reverses a non-negative square
Find principal square roots and distinguish them from the solutions of a squared equation.
The principal square root √A is the non-negative number whose square is A. Because 7² = 49, √49 = 7. A square of area 49 cm² has side length 7 cm. The area has square units; the root returns to length units. Also √0 = 0.
Explore the relationship
Predict first. Change one value, then explain the result. Diagrams show the entered dimensions; displayed decimals are rounded.
A radical and an equation ask different questions
√64 equals 8, not ±8. In contrast, solving x² = 64 gives x = 8 or x = −8, because both numbers square to 64. In a side-length problem only the positive solution is meaningful. For a negative radicand there is no real square root: squaring any real number gives a non-negative value.
Use number relationships rather than memorizing alone
Knowing 12² = 144 tells you √144 = 12. It also helps you recognize related decimal values: √1.44 = 1.2 because 1.2 × 1.2 = 1.44. A non-perfect-square positive integer has an irrational square root; its decimal expansion neither terminates nor repeats. This claim does not apply to every non-integer input.
Worked examples
Evaluate √0.81 and solve x² = 0.81
- 0.9 × 0.9 = 0.81, so √0.81 = 0.9.
- The equation x² = 0.81 has solutions 0.9 and −0.9.
- The radical itself names only the non-negative root.
Guided Practice
What is √121? How is that different from solving x² = 121?
A first hint
Ask whether the notation requests one principal root or all solutions.
Compare your reasoning
√121 = 11. The equation has solutions x = 11 and x = −11.
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
Compare √(36 + 64) with √36 + √64. Explain why a square root cannot generally be distributed over addition.
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