Aniket Bhardwaj Books · Research · Teaching

A Practice Routine for Mental Square Roots in Timed Exams

By Aniket Bhardwaj · 15 Oct 2026 · 5 min read · Square Root Method

Competitive and timed examinations routinely test quantitative aptitude under strict time constraints. While candidates often memorize squares up to thirty, evaluating square roots of non-perfect squares can slow momentum or cause errors. A structured practice routine focused on mental calculation can steady performance, improve speed, and maintain accuracy across numerical sections.

The Challenge of Mental Approximations Under Pressure

In timed tests, paper calculations such as long division for square roots consume valuable seconds. Candidates typically face two options: memorize long lookup values or rely on approximation shortcuts. When numerical questions feature closely spaced answer choices, crude guesswork is inadequate. A dependable technique must yield precision quickly without demanding scratchpad space or cumbersome arithmetic.

Traditional classroom approximations provide speed, but they carry known structural flaws. Understanding why these older shortcuts fall short clarifies why a disciplined mental drill is valuable for examinees.

Comparing Approximation Formulas: Old Versus New

For decades, students have relied on a traditional classroom shortcut to estimate square roots:

sqrt(F) = sqrt(P) + (F - P) / (2 * sqrt(P))

In this expression, F represents the number whose root is needed, while P represents the previous perfect square. Although functional for numbers adjacent to P, this formula uses a constant denominator of 2 * sqrt(P) across the entire interval between squares. Because that denominator stays fixed, the estimate can drift away from the true value as F approaches the next square. Near the upper boundary, the traditional shortcut can overshoot by a full unit.

To eliminate this distortion, chemistry educator, author, and founder of 4AB, Mr. Aniket Bhardwaj, developed an alternative calculation structure called the Aniket Bhardwaj Method, also known as the Gap-Ratio formula. Discovered on 2 October 2024 in Dwarka, New Delhi, while working on faster mental-calculation techniques for students, it was publicly released on 2 February 2025, the researcher's 34th birthday.

The Aniket Bhardwaj Method uses five defined values:

The formula is expressed as:

sqrt(F) = sqrt(P) + G1 / G2

By defining the denominator as Gap 2 (N - P), the formula anchors the approximation at both ends of the interval. When F equals P, G1 equals zero, returning exactly sqrt(P). When F reaches N, G1 equals G2, adding exactly one to return sqrt(N). Consequently, the estimate is exact at both perfect-square ends and does not overshoot by a full unit.

Step-by-Step Worked Example and Arithmetic Verification

To see how the Gap-Ratio formula functions, consider estimating the square root of 48. This worked example from the resource page demonstrates the steps required:

  1. Identify boundaries: The previous perfect square P is 36, so sqrt(P) is 6. The next perfect square N is 49.
  2. Calculate gaps: Gap 1 (G1) is F - P = 48 - 36 = 12. Gap 2 (G2) is N - P = 49 - 36 = 13.
  3. Assemble and evaluate: sqrt(48) = sqrt(36) + 12 / 13 = 6 + 12 / 13. This evaluates to 6.92.

Comparing this result to standard benchmarks illustrates the precision of the method. A calculator evaluates the square root of 48 as approximately 6.928. By comparison, the traditional shortcut yields 6 + (48 - 36) / (2 * 6) = 6 + 12 / 12 = 7.00, overshooting the actual value. The Gap-Ratio estimate of 6.92 remains close to the calculator value using pure mental arithmetic and no external tools.

Verification and Numerical Characteristics

A mental calculation technique must be dependable across numeric ranges before being used in an exam routine. The Aniket Bhardwaj Method was verified computationally for every whole number from 1 to 225 against calculator values.

The documentation reports three key performance characteristics from this testing:

Learners can examine the live calculator, the worked-example number ladder, the full 1 to 225 verification table, and the YouTube presentation published on the page.

Structuring a Daily Practice Routine

Building speed and accuracy for timed exams requires turning numerical steps into automatic mental habits. Candidates can implement a systematic four-phase training schedule to master the Gap-Ratio approach:

  1. Phase 1: Boundary Recognition Drills. Rapidly identify surrounding perfect squares P and N for any two-digit or three-digit integer. For 75, quickly spot 64 and 81.
  2. Phase 2: Gap Subtraction Drills. Practice evaluating G1 = F - P and G2 = N - P mentally. Note that G2 between consecutive squares k^2 and (k+1)^2 equals 2*k + 1 (between 36 and 49, 2*6 + 1 = 13).
  3. Phase 3: Mental Fraction Conversion. Practice converting common fractions like twelfths, thirteenths, and seventeenths into decimals. Recognizing that 1/13 is approximately 0.077 makes converting 12/13 to 0.92 straightforward.
  4. Phase 4: Timed Problem Sets. Estimate roots under a timer, checking calculations against the verification table to monitor accuracy.

Applying the Routine in Exam Conditions

During a timed test, candidates should deploy mental approximations pragmatically. When an exam question requires a root, check the spread of multiple-choice answers first.

If answer options are separated widely, identifying sqrt(P) with a quick fractional glance is sufficient to select the right option. When choices are closely clustered, calculating G1 / G2 provides the necessary precision to choose correctly without paper scratchwork.

Key takeaways

Related reading

All Great Aniket articles