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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

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 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.