MINDMAPS FOR UPSCPET 2026 · VISUAL REVISION
Powers, roots and square arrangements
Use exponent laws, square checks and nearest perfect squares
Powers, roots and square grids
Laws of exponents
- For integer m,n and a ≠ 0: aᵐ × aⁿ = aᵐ⁺ⁿ and aᵐ ÷ aⁿ = aᵐ⁻ⁿ.
- Power of a power or product: (aᵐ)ⁿ = aᵐⁿ and (ab)ᵐ = aᵐbᵐ.
- In 3⁴ × 3ˣ = 3⁸, matching exponents gives 4 + x = 8, so x = 4.
Exact decimal square roots
- When a terminating decimal is a perfect square, each pair of decimal digits gives one root digit.
- Always verify decimal placement by squaring: 1.5² = 2.25, 0.08² = 0.0064, 1.7² = 2.89.
- Adding the verified roots: 1.5 + 0.08 + 1.7 + 1 = 4.28.
Grouping cubes before dividing
- Rewrite constants as powers before taking roots; for example, 8 = 2³.
- Combine bases under one exponent: (9³ × 6³)/2³ = ((9 × 6)/2)³ = 27³.
- For x³ = 27³ with x > 0, the real cube root gives x = 27 directly.
Largest solid square grid
- An n × n square grid uses n² items; the largest possible side is floor(√N).
- For 1000 items, 31² = 961 and 32² = 1024 > 1000, leaving 1000 − 961 = 39 unused.
- For 1250 items, 35² = 1225 and 36² = 1296 > 1250, leaving 1250 − 1225 = 25 unused.
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