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Square Root of 78 — Value, Simplification, and Steps

Algebra

TL;DR

The square root of 78 is irrational and approximately 8.832, and √78 is already in simplest radical form because 78 has no perfect-square factor. This article shows the estimation and long-division methods, gives a quick-reference table, and points to where √78 appears.

Last updated on: July 18, 2026 5 min read

The square root of 78 is approximately 8.832 (irrational, so it never terminates or repeats). Because 78 has no square factor, √78 cannot be simplified further.

Quick Answer:
Result: √78 ≈ 8.832
Notation: irrational decimal, non-terminating
Method shown: long division and estimation
Approximate value: 8.832 (to 3 decimal places)
Exact form: √78, already in simplest radical form (78=2×3×13, no square factor)

Quick Reference Table

Where the Square Root of 78 Appears

√78 is the length of the diagonal of a box whose squared edge-lengths add to 78, for instance, a rectangle with sides √13 and √65, since 13 + 65 = 78. It also appears as the distance (√{(x_2 - x_1)^2 + (y_2 - y_1)^2}) between two points whose coordinate differences square to 78, such as the points (0,0) and (7,√29).

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=8² and 81=9², 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. That same factorisation is why √78 is already in simplest radical form.

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: √78 is already in simplest radical form.

Method 2: Estimation

The nearest perfect squares are 64=8² and 81=9², so √78 lies between 8 and 9.

Since 78 is much closer to 81 than to 64, the root is close to 9.

Test 8.82=77.44 and 8.92=79.21, so the root is between 8.8 and 8.9.

Final answer: √78 ≈ 8.83.

Method 3: Long Division

Pair the digits from the right and add decimal pairs: 78.0000.

Largest square ≤78 is 64=8²; first digit 8, remainder 14.

Bring down 000000 to make 1400; double the 8 to get 16, and find d so that 16d×d≤1400: 168×8=1344, so the next digit is 8, remainder 56.

Continue: the next digits give 8.832….

Final answer: √78 ≈ 8.832.

Common Mistakes With Square Root of 78

Mistake 1: Trying to simplify a radical with no square factor

Where it slips in: assuming every radical reduces to something smaller. Don't do this: write 78=3√26 or 2392 as if a factor came out. The correct way: check the prime factors first.

Mistake 2: Rounding too early

Where it slips in: writing 78=8.8 and treating it as exact. Don't do this: report 8.8 in a calculation that needs precision. The correct way: keep the exact form through the algebra and round only at the final step.

Mistake 3: Confusing 78 with a nearby perfect square

Where it slips in: expecting a clean whole-number answer. Don't do this: guess 78=8. The correct way: understand its root is irrational, sitting between 8 and 9.

Where to Go From Here

Estimate √75 and √80 using the same methods above.

Frequently Asked Questions

1. Is the square root of 78 rational or irrational?
   Irrational. Prime factors have no repeated pairs.
2. Can the square root of 78 be simplified?
   No. Already in simplest radical form.
3. What is the square root of 78 to two decimal places?
   Approximately 8.83.
4. What is the square root of 81?
   9, exactly.
5. What is the square root of -78?
   There is no real square root; it involves imaginary numbers.