Square Root of 640 — Value, Simplification, and Steps

What Is the Square Root of 640?

The square root of 640 is $\sqrt{640} = 8\sqrt{10} \approx 25.298$. It is irrational: the decimal continues without repeating, but the exact simplified radical is the clean $8\sqrt{10}$.

Quick Answer:

Result: $\sqrt{640} = 8\sqrt{10} \approx 25.298$

Notation: $\sqrt{640}$, simplified $8\sqrt{10}$, or $\approx 25.298$

Method shown: Prime factorization to simplify, long division for the decimal

Rational or irrational: Irrational (640 is not a perfect square)

Exact form: $8\sqrt{10}$

Since $25^2 = 625$ and $26^2 = 676$, the root of 640 falls between 25 and 26, just above 25.

Quick Reference Table

The table shows how nearby roots simplify. A number simplifies only when it carries a perfect-square factor; 640 carries the factor 64.

Where the Square Root of 640 Appears

$\sqrt{640}$ shows up when an area of 640 square units must be turned into the side of a square, giving a side of $8\sqrt{10} \approx 25.298$ units. It also appears in geometry and physics whenever a squared quantity works out to 640, where keeping the exact $8\sqrt{10}$ form avoids rounding until the final decimal is needed.

What Does "Simplifying a Square Root" Mean?

Simplifying a square root means pulling every perfect-square factor out from under the radical so the number inside is as small as possible. The square root of a product splits as $\sqrt{ab} = \sqrt{a},\sqrt{b}$.

A perfect square is an integer times itself, like $64 = 8^2$. Because $640 = 64 \times 10$ and 64 is a perfect square, the 8 comes out and 10 stays in, giving $8\sqrt{10}$.

Is the square root of 640 rational or irrational? It is irrational. The leftover factor 10 is not a perfect square, so $\sqrt{10}$ never terminates, which keeps $8\sqrt{10}$ irrational, the same as √47. Only a perfect square such as √4900 gives a rational root.

How to Compute the Square Root of 640

Method 1: Prime factorization and simplification

Factor 640 into primes.

$640 = 2^7 \times 5$

Group the primes into pairs: $2^7 = 2^6 \times 2 = (2^3)^2 \times 2$.

$640 = (2^3)^2 \times 2 \times 5 = 64 \times 10$

Take the square root, pulling out the perfect square.

$\sqrt{640} = \sqrt{64} \times \sqrt{10} = 8\sqrt{10}$

Final answer: $\sqrt{640} = 8\sqrt{10}$

Method 2: Spotting the largest perfect-square factor

List perfect-square factors of 640: 4, 16, 64. The largest is 64.

$640 = 64 \times 10$

$\sqrt{640} = \sqrt{64},\sqrt{10} = 8\sqrt{10}$

Using a smaller factor still works but needs a second pass: $\sqrt{640} = \sqrt{16 \times 40} = 4\sqrt{40} = 4 \times 2\sqrt{10} = 8\sqrt{10}$.

Final answer: $\sqrt{640} = 8\sqrt{10}$

Method 3: Long division for the decimal

Pair digits as $\overline{6}\ \overline{40}.\overline{00}$.

Largest square under 6 is 4, so the first digit is 2; remainder $6 - 4 = 2$, bring down 40 to make 240.

Double 2 to get 4; $4d \times d \le 240$ needs $d = 5$, since $45 \times 5 = 225$.

Quotient 25; remainder $240 - 225 = 15$, bring down 00 to make 1500.

Double 25 to get 50; $50d \times d \le 1500$ needs $d = 2$, since $502 \times 2 = 1004$.

Quotient reads $25.2\ldots$, refining to $\approx 25.298$.

Final answer: $\sqrt{640} \approx 25.298$

Common Mistakes With the Square Root of 640

Mistake 1: Stopping at a smaller factor

Where it slips in: A student factors $640 = 16 \times 40$, writes $4\sqrt{40}$, and calls it simplified.

Don't do this: Leave $4\sqrt{40}$ as the answer. The 40 still hides a perfect-square factor of 4.

The correct way: Keep factoring until nothing square remains: $4\sqrt{40} = 4 \times 2\sqrt{10} = 8\sqrt{10}$. Using the largest square factor, 64, gets there in one step.

Mistake 2: Multiplying the outside and inside numbers

Where it slips in: Reading $8\sqrt{10}$, a student computes $8 \times 10 = 80$ under a single radical.

Don't do this: Rewrite $8\sqrt{10}$ as $\sqrt{80}$. The 8 is a coefficient, not a factor inside the root.

The correct way: To move 8 inside, square it: $8\sqrt{10} = \sqrt{64 \times 10} = \sqrt{640}$, which returns the original.

Mistake 3: Treating 640 as a perfect square

Where it slips in: The round-looking 640 tempts students to expect a whole-number root.

Don't do this: Guess 25 or 26 as an exact answer.

The correct way: Check the prime factorization. Since $2^7 \times 5$ has an unpaired 2 and a lone 5, the root is irrational, and the exact form is $8\sqrt{10}$.

Conclusion

• The square root of 640 is $8\sqrt{10}$, about 25.298, and it is irrational.

• $640 = 64 \times 10$, so the perfect square 64 leaves the radical as 8.

• Its prime factorization $2^7 \times 5$ has unpaired factors, confirming the root is not whole.

• Long division gives the decimal $\approx 25.298$ for when a number is needed.

• The common slips are stopping at a smaller factor, merging the coefficient into the radical, and expecting a whole-number root.

For step-by-step help simplifying radicals with a teacher, explore Bhanzu's algebra tutor or browse math classes online.

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