How to Calculate Square Roots
Learn how to calculate square roots manually and understand their properties. Covers perfect squares, simplifying radicals, the long division method, and real-world uses of square roots.
What Is a Square Root?
The square root of a number n is a value x such that x² = n, written as x = √n. Every positive number has two square roots: one positive (the principal root) and one negative. For example, √25 = 5 because 5² = 25, and −5 is also a square root since (−5)² = 25.
Perfect Squares
A perfect square is a number that has an integer as its square root. Common perfect squares include 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, and 144. Recognizing perfect squares makes it faster to simplify radicals and estimate square roots of nearby numbers.
Estimating Square Roots of Non-Perfect Squares
To estimate √50, identify the two perfect squares it falls between: 49 and 64. Since √49 = 7 and √64 = 8, √50 is between 7 and 8, and closer to 7. A more precise estimate uses linear interpolation: √50 ≈ 7 + (50 − 49)/(64 − 49) × 1 ≈ 7.07.
Simplifying Radical Expressions
To simplify a radical, factor out perfect square factors from under the square root sign. For √72, factor as √(36 × 2) = √36 × √2 = 6√2. Always look for the largest perfect square factor to complete the simplification in one step. The result 6√2 is the simplified radical form of √72.
The Long Division Method for Square Roots
The long division method allows you to calculate square roots by hand to any desired precision. Group the digits in pairs from the decimal point outward. Find the largest integer whose square is ≤ the first group, subtract, bring down the next pair, and repeat. This method is systematic and works for any number, though a calculator is more practical for most purposes.
Square Roots and the Pythagorean Theorem
Square roots are essential for applying the Pythagorean theorem in geometry: c = √(a² + b²). For a right triangle with legs 5 and 12, the hypotenuse is √(25 + 144) = √169 = 13. Square roots also appear in the distance formula, quadratic formula, and standard deviation calculation.
Properties of Square Roots
Key properties include: √(a × b) = √a × √b, and √(a/b) = √a / √b. These allow you to split or combine radicals when simplifying. However, √(a + b) ≠ √a + √b — this is one of the most common errors students make. For example, √(9 + 16) = √25 = 5, not √9 + √16 = 3 + 4 = 7.
Try These Calculators
Put what you learned into practice with these free calculators.
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