Square Root of 260 — Value, Simplification, and Steps
Square Root of 260 — Value, Simplification, and Steps
TL;DR
The square root of 260 is (2 \sqrt{65}) in simplest radical form and approximately 16.125 as a decimal. This article shows how to simplify (\sqrt{260}) by prime factorization, estimate it between whole numbers, and explains why (\sqrt{260}) is an irrational number.
The square root of 78 is approximately 8.832 (irrational, so it never terminates or repeats). Because 78 has no square factor, (\sqrt{78}) cannot be simplified further.
Quick Answer:
Result: (\sqrt{78} \approx 8.832)
Notation: irrational decimal, non-terminating
Method shown: long division and estimation
Approximate value: 8.832 (to 3 decimal places)
Exact form: (\sqrt{78}), already in simplest radical form (78=2×3×13)
Quick Reference Table
| Expression | Approx. Value | Perfect Square? |
|---|---|---|
| (\sqrt{64}) | 8.000 | Yes |
| (\sqrt{75}) | 8.660 | No |
| (\sqrt{78}) | 8.832 | No |
| (\sqrt{80}) | 8.944 | No |
| (\sqrt{81}) | 9.000 | Yes |
| 782 | 6,084 | — |
Where the Square Root of 78 Appears
(\sqrt{78} \approx 8.832) is the length of the diagonal of a box whose squared edge-lengths add to 78, for instance a rectangle with sides (\sqrt{13}) and (\sqrt{65}). It also appears as the distance between two points whose coordinate differences square to 78.
What a Square Root Is
A square root of a number n is a value that, multiplied by itself, gives n. When n is a perfect square, the root is a whole number; when it is not, the root is irrational, a decimal that runs forever without repeating. Because 78 sits between the perfect squares 64 and 81, its square root falls between 8 and 9.
Is the Square Root of 78 Rational or Irrational?
The square root of 78 is irrational. Its prime factorisation is 78=2×3×13, three distinct primes with no repeated pair, so no whole number squares to 78 and the decimal never terminates.
How to Compute the Square Root of 78
Method 1: Simplify the Radical (Check First)
List the prime factors of 78.
78=2×3×13
No prime appears twice, so there is no square factor to pull out.
Final answer: (\sqrt{78}) is already in simplest radical form.
Method 2: Estimation
The nearest perfect squares are 64 and 81, so (\sqrt{78}) lies between 8 and 9.
Final answer: (\sqrt{78} \approx 8.83).
Method 3: Long Division
Pair the digits from the right and add decimal pairs: 78.0000. The largest square ≤78 is 64; the next digits give 8.832…
Final answer: (\sqrt{78} \approx 8.832).
Common Mistakes With Square Root of 260
Mistake 1: Trying to simplify a radical with no square factor
Where it slips in: assuming every radical reduces to something smaller. The correct way: check the prime factors first.
Mistake 2: Rounding too early
Where it slips in: writing (\sqrt{78} = 8.8) and treating it as exact. The correct way: keep the exact form through the algebra.
Mistake 3: Confusing 78 with a nearby perfect square
Where it slips in: expecting a clean whole-number answer. The correct way: recognize that 78 is not a perfect square, so its root is irrational.
Where to Go From Here
Estimate (\sqrt{75}) and (\sqrt{80}) by the same method, then check your guesses against the table above. To build these skills with a teacher, explore Bhanzu's online math classes.
Frequently Asked Questions
What is the square root of 260 in simplest radical form?
It is (2 \sqrt{65}).
Is the square root of 260 rational or irrational?
Irrational. The root cannot be expressed as a fraction.
What is (\sqrt{260}) as a decimal?
Approximately 16.125.
Can (\sqrt{260}) be simplified further than (2\sqrt{65})?
No. The remaining radicand 65 has no repeated prime factor, so it is fully simplified.