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Determine square roots of rational numbers

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Square Roots of Rational Numbers

To find the square root of a rational number written as a fraction, take the square root of the numerator and denominator separately: the square root of a over b equals the square root of a over the square root of b. Learn the rule, a worked example, and when the result stays a rational number.

The square root of a fraction rule

To find the square root of a rational number written as a fraction, you can take the square root of the numerator and the denominator separately: √(a/b) = √a / √b, as long as b is not zero. This works because squaring √a/√b gives back a/b exactly — the result is itself a rational number you can compare and order against others.

Square root of a rational number (fraction) Rule: the square root of a/b equals the square root of a divided by the square root of b, provided b is not zero. Worked example: the square root of 4/9 equals the square root of 4 over the square root of 9, which is 2/3. a b = a / b square root of a fraction 4 9 = 4 / 9 = 2/3
√(a/b) = √a ÷ √b. Worked example: √(4/9) = √4 ÷ √9 = 2/3.

Worked example

To find √(4/9), split it into √4 over √9. Since √4 = 2 and √9 = 3, the answer is 2/3. Checking: (2/3)² = 4/9, confirming the result.

When the result stays rational

The result is a rational number only when both √a and √b are whole numbers — that is, when the numerator and denominator are both perfect squares. If either is not a perfect square, the square root of that fraction is irrational, and you would leave it in simplified radical form instead.

Simplifying before taking the root

If a fraction is not yet in simplest form, simplifying it first can reveal a perfect-square numerator and denominator that weren't obvious. For example, √(8/50) simplifies to √(4/25) = 2/5, which is much easier to evaluate directly. When neither part is a perfect square, you can still estimate the square root instead of leaving it unevaluated.

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