What Is a Square Root?
A square root of a number is a value that multiplies by itself to give that number. Squaring and taking the square root are inverse operations, so $80 \times 80 = 6400$ and $\sqrt{6400} = 80$.
A perfect square is a number that is the square of a whole number. 6400 qualifies because $80^2 = 6400$, which is why its root leaves no radical behind. The wider set of exact roots is collected in the guide to squares and square roots.
Where Does the Square Root of 6400 Appear?
$\sqrt{6400}$ appears any time an area of 6400 square units belongs to a square, since that square has sides of length 80. It also shows up in scaling and unit work, for example a plot of 6400 square metres laid out as an $80 \times 80$ square, or the side of a screen region holding 6400 pixels arranged squarely. Because the root is a clean whole number, 6400 is a common textbook value for perfect-square practice
Quick Reference Table
The table places 6400 among nearby perfect squares. Because 6400 is a perfect square, its root is exact.
Number | Square Root | Type |
|---|---|---|
$\sqrt{6241}$ | $79$ | Rational (perfect square) |
$\sqrt{6400}$ | $80$ | Rational (perfect square) |
$\sqrt{6561}$ | $81$ | Rational (perfect square) |
$\sqrt{64}$ | $8$ | Rational (perfect square) |
$\sqrt{100}$ | $10$ | Rational (perfect square) |
$\sqrt{640}$ | $\approx 25.298$ | Irrational |
$\sqrt{6500}$ | $\approx 80.623$ | Irrational. |
Is the Square Root of 6400 Rational or Irrational?
$\sqrt{6400}$ is rational. It equals the whole number 80, which is the ratio $\tfrac{80}{1}$, so it fits the definition of a rational number exactly.
A whole number has a rational square root only when it is a perfect square. 6400 is one, since its prime factorisation pairs perfectly, so unlike an irrational root such as $\sqrt{6500}$, the value terminates cleanly at 80.
How Do You Find the Square Root of 6400?
Method 1: Prime factorization
Break 6400 into prime factors.
$6400 = 64 \times 100$
$64 = 2^6$ and $100 = 2^2 \times 5^2$, so $6400 = 2^8 \times 5^2$.
Take one factor from each pair: $\sqrt{6400} = 2^4 \times 5$.
$2^4 \times 5 = 16 \times 5 = 80$.
Final answer: $\sqrt{6400} = 80$
Method 2: Split into known roots
Write 6400 as a product of two perfect squares.
$6400 = 64 \times 100$
$\sqrt{6400} = \sqrt{64} \times \sqrt{100}$
$\sqrt{64} = 8$ and $\sqrt{100} = 10$.
$8 \times 10 = 80$.
Final answer: $\sqrt{6400} = 80$
Method 3: Long division (confirmation)
Pair the digits of 6400 as $\overline{64}\ \overline{00}$.
The largest square under 64 is 64 itself, and $\sqrt{64} = 8$, so the first digit is 8; remainder $64 - 64 = 0$.
Bring down $00$ to make 0, then double 8 to get 16, and find $d$ with $(160 + d) \times d \le 0$: $d = 0$.
The quotient is 80 with remainder 0, which confirms an exact perfect square.
Final answer: $\sqrt{6400} = 80$
Examples of the Square Root of 6400
Example 1: Where Students Lose the Mark
A student reasons that $\sqrt{64} = 8$, then "adds the two zeros" and writes $\sqrt{6400} = 800$.
The zeros come from the factor 100, whose root is 10, not 100.
$\sqrt{6400} = \sqrt{64} \times \sqrt{100} = 8 \times 10$.
So $\sqrt{6400} = 80$, not 800.
Example 2: Verifying by Squaring
Check that 80 is the correct root.
Square the candidate answer.
$80 \times 80 = 6400$.
The product matches, so $\sqrt{6400} = 80$.
Example 3: Solving x² = 6400
Solve $x^2 = 6400$ for $x$.
Take the square root of both sides.
$x = \pm 80$.
Both $80$ and $-80$ square to 6400.
Example 4: Simplifying √6400 as a Radical
Show that no radical remains.
$6400 = 2^8 \times 5^2$, and every prime appears an even number of times.
Every factor comes out: $\sqrt{6400} = 2^4 \times 5 = 80$.
The simplest form is the whole number 80.
Example 5: Scaling a Related Root
Use 6400 to find $\sqrt{16 \times 6400}$.
$\sqrt{16 \times 6400} = \sqrt{16} \times \sqrt{6400}$.
$\sqrt{16} = 4$ and $\sqrt{6400} = 80$.
$4 \times 80 = 320$.
Common Mistakes
Mistake 1: Misplacing the Zeros
Where it slips in: A student roots the digit block and then reattaches the wrong number of zeros.
Don't do this: Write $\sqrt{6400} = 800$ by keeping both zeros after rooting 64.
The correct way: Root the perfect-square factors separately: $\sqrt{64} \times \sqrt{100} = 8 \times 10 = 80$.
Mistake 2: Assuming It Is Irrational
Where it slips in: A large four-digit number looks like it should have a messy decimal root.
Don't do this: Reach for a calculator estimate and report $\sqrt{6400} \approx 80.0$ as if it were only approximate.
The correct way: Check whether the number is a perfect square first. 6400 is $80^2$, so the root is exactly 80.
Mistake 3: Dropping the Negative Root When Solving
Where it slips in: Solving the equation $x^2 = 6400$ rather than evaluating the principal root.
Don't do this: Give only $x = 80$ and forget the other solution.
The correct way: The principal root $\sqrt{6400}$ is 80, but $x^2 = 6400$ has two solutions, $x = \pm 80$.
Conclusion
The square root of 6400 is exactly 80 and is rational.
6400 is a perfect square, $80^2$, so no radical remains.
Prime factorization ($2^8 \times 5^2$) and splitting into $\sqrt{64} \times \sqrt{100}$ both give 80.
Long division confirms the result with remainder zero.
The equation $x^2 = 6400$ has two roots, $x = \pm 80$, but $\sqrt{6400}$ alone is $+80$.
To build perfect-square fluency with a teacher, explore Bhanzu's algebra tutor sessions or browse math classes online. Want a live Bhanzu trainer to walk through more square-root problems? Book a free demo class.
For a plain-language reference on the operation, see Math is Fun on squares and square roots.
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