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# Square Root of 640 — Value, Simplification, and Steps

[Algebra](/content/tag/algebra/index.html)

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.

BT

[Bhanzu Team](/content/authors/bhanzu-team/index.html) 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.

| Number     | Simplified form | Approximate value  |
|------------|------------------|--------------------|
| \(\sqrt{625}\) | \(25\)           | 25 (exact, perfect square) |
| \(\sqrt{640}\) | \(8\sqrt{10}\)  | ≈25.298            |
| \(\sqrt{648}\) | \(18\sqrt{2}\)  | ≈25.456            |
| \(\sqrt{160}\) | \(4\sqrt{10}\)  | ≈12.649            |
| \(\sqrt{10}\)  | \(\sqrt{10}\)    | ≈3.162             |
| \(\sqrt{676}\) | \(26\)           | 26 (exact, perfect square) |

## 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](/content/math/terms/square-root/index.html) 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.