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
Expression | Approx. Value | Perfect Square? |
|---|---|---|
$\sqrt{64}$ | $8.000$ | Yes ($8^2$) |
$\sqrt{75}$ | $8.660$ | No ($= 5\sqrt{3}$) |
$\sqrt{78}$ | $8.832$ | No (simplest form) |
$\sqrt{80}$ | $8.944$ | No ($= 4\sqrt{5}$) |
$\sqrt{81}$ | $9.000$ | Yes ($9^2$) |
$78^2$ | $6{,}084$ | — |
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 $\sqrt{65}$, 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 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 $\sqrt{78} = 3\sqrt{26}$ 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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