Is 400 a Perfect Square? Yes — √400 = 20

Perfect Square Checker

Use the checker above to determine whether any non-negative integer is a perfect square. A perfect square is a non-negative integer that can be written as the product of an integer with itself: n = k² for some integer k. The first perfect squares are 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, ...

Step-by-Step: Why 400 Is a Perfect Square

1. Compute √400:
   √400 = 20 (an integer).
2. Verify by squaring:
   20 × 20 = 400  ✓
3. Prime factorization check:
   400 = 24 × 52
   All exponents are even ⇒ 400 is a perfect square.

√400 = 20

⚡ Why this matters: perfect squares are the only positive integers with an odd number of divisors. 400 has 15 divisors — an odd count, confirming 400 is a perfect square. See all divisors of 400.

What Is a Perfect Square?

A perfect square is a non-negative integer that is the square of an integer:

n is a perfect square  ⇔  n = k²,  k ∈ ℤ_≥0

Equivalently, n is a perfect square if and only if √n is an integer. Geometrically, you can arrange n identical unit squares into a square grid only if n is a perfect square.

Prime-factorization rule: n is a perfect square iff every prime in the prime factorization of n appears with an even exponent. For example, 144 = 2⁴ × 3² (both even) is a perfect square; 72 = 2³ × 3² has an odd exponent on 2, so it is not.

Nearby Examples

Related Operations

• Square Root of 400 — decimal value with Babylonian Method
• What is 400 Squared? — compute n²

References

• [1] Square number (Wikipedia)
• [2] Square Number (Wolfram MathWorld)

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