Exact root
- Input
- √225
- Expected output
- 15
15 × 15 = 225, a perfect square, so the result is a whole number, not an approximation.
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square root
The square root of x is the number that, squared, gives back x. When x is a perfect square the result is an integer; in most other cases it is a never-ending decimal that the calculator can only display up to a certain number of digits.
15 × 15 = 225, a perfect square, so the result is a whole number, not an approximation.
10 is not a perfect square, so √10 is irrational; Heron's method starting from 3 already converges to this value in 3 iterations.
No real number squared is negative, so −16 has no real square root; the cube root of −27, in contrast, is −3.
A negative exponent denotes the reciprocal of the positive power: a⁻ⁿ = 1 ÷ aⁿ. For example, 2⁻³ = 1 ÷ 2³ = 1 ÷ 8 = 0.125.
There is no real square root, because squaring any real number, positive or negative, never gives a negative result; the calculator flags this case instead of returning a value.
Use Heron's method: start from a guess x₀ and refine with xₙ₊₁ = (xₙ + N/xₙ) ÷ 2; for √10 starting at x₀ = 3, three iterations already reach 3.16227766, with five correct decimal places.
An exact root happens when the radicand is a perfect square or cube, like √225 = 15 or the cube root of 125 = 5, with no remainder; an approximate root is the case for irrational numbers, like √10 ≈ 3.16227766017, an endless decimal the calculator truncates at 12 significant digits.
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