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# Square Root of 68 — Value & Simplified Form (2√17)

## TL;DR
The square root of 68 is \( 2\sqrt{17} \) in simplified radical form and approximately 8.246 as a decimal. This article gives the exact value, the prime-factorisation and long-division methods, a quick-reference table, and where \( \sqrt{68} \) appears.

## The square root of 68
The square root of 68 is **\( 2\sqrt{17} \approx 8.246 \)**. It is an irrational number, so its decimal never terminates or repeats.

> **Quick Answer:**
>
> **Result:** \( \sqrt{68} = 2\sqrt{17} \)
>
> **Notation:** simplified radical \( 2\sqrt{17} \); decimal \( \approx 8.246 \)
>
> **Method shown:** prime factorisation (for the radical) and long division (for the decimal)
>
> **Approximate value:** 8.246 (to 3 decimal places)
>
> **Exact form:** \( 2\sqrt{17} \) (not a whole number — 68 is not a perfect square)

## Quick Reference Table

| Expression | Simplified form | Decimal (3 dp) |
| --- | --- | --- |
| \( \sqrt{64} \) | 8 | 8.000 |
| \( \sqrt{65} \) | \( \sqrt{65} \) | 8.062 |
| \( \sqrt{68} \) | \( 2\sqrt{17} \) | 8.246 |
| \( \sqrt{72} \) | \( 6\sqrt{2} \) | 8.485 |
| \( \sqrt{17} \) | \( \sqrt{17} \) | 4.123 |
| 682 | — | 4624 |

## Where the Square Root of 68 Appears

\( \sqrt{68} \) is the length of the diagonal of a rectangle with sides 2 and 8, since \( 2^2 + 8^2 = 4 + 64 = 68 \). It also turns up as the distance between the points (0,0) and (2,8) on a coordinate plane, because the distance formula reduces to the same square root.

## What a Square Root Is

The **square root** of a number is the value that, multiplied by itself, gives that number. Because \( 8^2 = 64 \) and \( 9^2 = 81 \), the square root of 68 sits between 8 and 9. Since 68 is not a perfect square, its root is irrational — a full explanation lives on the [square root](/content/math/terms/square-root/index.html) page.

## How to Compute the Square Root of 68

### Method 1: Prime factorisation (for the simplified radical)

Break 68 into primes.

68 = 2 × 34 = 2 × 2 × 17 = \( 2^2 \times 17 \)

Take the pair of 2s outside the radical as a single 2:

\( \sqrt{68} = \sqrt{2^2 \times 17} = 2\sqrt{17} \).

**Final answer:** \( \sqrt{68} = 2\sqrt{17} \).

### Method 2: Estimation (for a quick decimal)

Find the two perfect squares 68 sits between.

\( 8^2 = 64 \) and \( 9^2 = 81 \), so \( \sqrt{68} \) is between 8 and 9.

Since 68 is much closer to 64 than to 81, the value is a little above 8.2.

**Estimate:** about 8.25, which matches the true value 8.246.

### Method 3: Long division (for the decimal to 3 dp)

Pair the digits and take the largest square ≤68, which is 64 = 8^2; the first digit is 8, remainder 4.

Bring down 000000: divisor becomes 16_, and 16 × 2 = 32 ≤ 40, so the next digit is 2; remainder 76.

Continue: the next digit is 4, then 6.

**Final answer:** \( 68 \approx 8.246 \).

## Common Mistakes With Square Root of 68

### **Mistake 1: Simplifying to the wrong factor**

**Where it slips in:** splitting 68 as 4 × 17 but forgetting that only 4 comes out. **Don't do this:** write \( \sqrt{68} = 4\sqrt{17} \). **The correct way:** \( 4 = 2 \), so \( \sqrt{68} = 2\sqrt{17} \), not 4.

### **Mistake 2: Calling 2√17 the decimal answer**

**Where it slips in:** stopping at the radical when a decimal is asked for. **Don't do this:** report \( \sqrt{68} = 2 × 17 = 34 \). **The correct way:** \( 2 = 4.123 = 8.246 \).

### **Mistake 3: Treating √68 as rational**

**Where it slips in:** rounding early and calling 8.25 exact. **Don't do this:** write \( \sqrt{68} = 8.25 \). **The correct way:** \( \, \sqrt{68} \) is irrational; 8.246 is a rounded approximation.

## Frequently Asked Questions

### Is the square root of 68 rational or irrational?

Irrational. 68 is not a perfect square, so its decimal (8.246…) never terminates or repeats.

### What is √68 in simplest radical form?

\( 2\sqrt{17} \). The prime factorisation \( 68 = 2^2 \times 17 \) lets one pair of 2s come out of the radical.

### What is the square root of 68 to two decimal places?

8.25 (rounded). To three places it is 8.246.

### Is √68 close to any whole number?

It sits between 8 and 9, closer to 8 — the value 8.246 rounds to 8.