Square Root of 260 — Value, Simplification, and Steps

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 ($78 = 2 \times 3 \times 13$, no square factor)

Quick Reference Table

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 $\sqrt{65}$, since $13 + 65 = 78$. It also appears as the distance $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$ between two points whose coordinate differences square to 78, such as the points $(0,0)$ and $(7, \sqrt{29})$.

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 $64 = 8^2$ and $81 = 9^2$, its square root falls between 8 and 9, unlike a whole-number root such as $\sqrt{441} = 21$.

Is the Square Root of 78 Rational or Irrational?

The square root of 78 is irrational. Its prime factorisation is $78 = 2 \times 3 \times 13$, three distinct primes with no repeated pair, so no whole number squares to 78 and the decimal never terminates. That same factorisation is why $\sqrt{78}$ 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.

$78 = 2 \times 3 \times 13$

No prime appears twice, so there is no square factor to pull out.

$\sqrt{78} = \sqrt{2 \times 3 \times 13}$

Final answer: $\sqrt{78}$ is already in simplest radical form.

Method 2: Estimation

The nearest perfect squares are $64 = 8^2$ and $81 = 9^2$, so $\sqrt{78}$ lies between 8 and 9.

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

Test $8.8^2 = 77.44$ and $8.9^2 = 79.21$, so the root is between 8.8 and 8.9.

Final answer: $\sqrt{78} \approx 8.83$.

Method 3: Long Division

Pair the digits from the right and add decimal pairs: $78 . 00\ 00$.

Largest square $\le 78$ is $64 = 8^2$; first digit 8, remainder 14.

Bring down $00$ to make 1400; double the 8 to get 16, and find $d$ so that $16d \times d \le 1400$: $168 \times 8 = 1344$, so the next digit is 8, remainder 56.

Continue: the next digits give $8.832\ldots$

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. Don't do this: write $\sqrt{78} = 3\sqrt{26}$ or $2\sqrt{39}$ as if a factor came out. The correct way: check the prime factors first. Since $78 = 2 \times 3 \times 13$ has no repeated prime, $\sqrt{78}$ stays as it is.

Mistake 2: Rounding too early

Where it slips in: writing $\sqrt{78} = 8.8$ and treating it as exact. Don't do this: report 8.8 in a calculation that needs precision, since $8.8^2 = 77.44 \ne 78$. The correct way: keep the exact form $\sqrt{78}$ through the algebra and round to $8.832$ only at the final step.

Mistake 3: Confusing 78 with a nearby perfect square

Where it slips in: expecting a clean whole-number answer like the roots of 64 or 81. Don't do this: guess $\sqrt{78} = 8$ or $9$. The correct way: 78 is not a perfect square, so its root is the irrational $8.832\ldots$, sitting between 8 and 9.

Where to Go From Here

Estimate $\sqrt{75}$ and $\sqrt{80}$ by the same between-two-squares method, then check your guesses against the table above. To build these skills with a teacher, explore Bhanzu's algebra tutor or math classes online.

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