What is the square root of 400, and why? Explain how to find it without a calculator.

Step-by-step solution

1. 1

   Start from the definition. The square root of a number $N$ is the value that, multiplied by itself, gives $N$. So finding $\sqrt{400}$ means solving

   $k\times k=400.$

   That is the entire content of the question — everything below is just an efficient way to find $k$.

2. 2

   Factor 400 into primes and pair them up. Repeated division gives

   $400=4\times 100=2^{2}\times(2^{2}\times 5^{2})=2^{4}\times 5^{2}.$

   Every exponent is even, which is exactly the condition for $400$ to be a perfect square. Halving each exponent:

   $\sqrt{400}=2^{4/2}\times 5^{2/2}=2^{2}\times 5=20.$

   This method works for any perfect square and needs no guessing.

3. 3

   Check by multiplying back. Squaring the candidate:

   $20\times 20=400 ✓.$

   A second useful check is the trailing-zeros rule: $400$ has two trailing zeros, so its square root has one, and $\sqrt{4}=2$ gives the leading digit — hence $20$.

4. 4

   Note the sign convention. Both $20$ and $-20$ square to $400$, but the radical symbol $\sqrt{\ }$ denotes the principal (non-negative) root, so $\sqrt{400}=20$. The two-sign answer $x=\pm20$ belongs to the equation $x^{2}=400$, not to the expression $\sqrt{400}$ — a distinction worth keeping straight.

5. 5

   Sanity-check by bracketing. Since $19^{2}=361$ and $21^{2}=441$, the root must lie strictly between $19$ and $21$, and the only integer there is $20$ ✓. Bracketing like this is how you estimate roots of numbers that are not perfect squares, such as $\sqrt{410}\approx 20.25$.