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Square Root of 640 — Value, Simplification, and Steps

Algebra

TL;DR

The square root of 640 is (8\sqrt{10}), about 25.298, and it is irrational because 640 is not a perfect square. This article shows how to pull the perfect-square factor 64 out of the radical, compute the decimal by long division, and avoid the usual simplification slips.

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Bhanzu Team Last updated on July 20, 2026 5 min read

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} = 8\sqrt{10})
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 (\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=8². Because 640=64×10 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.

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 = (2^3)^2 \times 2).

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

Take the square root:

(\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)

Final answer: (\sqrt{640} = 8\sqrt{10})

Method 3: Long division for the decimal

Pair digits as (640.00).

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

Quotient reads 25.2… refining to ≈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=16×40) and writes (4\sqrt{40}) and calls it simplified.

The correct way: Keep factoring until nothing square remains: (\sqrt{40} = 4 \times 2\sqrt{10} = 8\sqrt{10}).

Mistake 2: Multiplying the outside and inside numbers

Where it slips in: A student computes (8\sqrt{10}) as 8×10.

The correct way: Keep them separate: (8\sqrt{10}) remains distinct.

Mistake 3: Treating 640 as a perfect square

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

The correct way: Check the prime factorization. Since it has an unpaired factor, the root is irrational.

Conclusion

• The square root of 640 is (8\sqrt{10}), about 25.298, and it is irrational.
• Long division gives the decimal (\approx 25.298) for when a number is needed.
• Common slips include stopping at a smaller factor, merging the coefficient into the radical, and expecting a whole-number root.