The square root of 384 is $8\sqrt{6} \approx 19.596$.
Quick Answer:
Result: $\sqrt{384} = 8\sqrt{6} \approx 19.596$
Notation: radical form $\sqrt{384}$; simplified radical $8\sqrt{6}$; exponent form $384^{1/2}$
Method shown: prime factorisation (to simplify) + long-division method (to estimate)
Approximate value (irrational): $19.59592$ (to 5 decimal places)
Exact form: $8\sqrt{6}$
Quick Reference Table
The table places $\sqrt{384}$ among nearby square roots and shows which are exact.
Number $n$ | Square root $\sqrt{n}$ | Exact or approximate |
|---|---|---|
361 | $19$ | Exact (perfect square) |
384 | $8\sqrt{6} \approx 19.596$ | Irrational |
400 | $20$ | Exact (perfect square) |
96 | $4\sqrt{6} \approx 9.798$ | Irrational |
150 | $5\sqrt{6} \approx 12.247$ | Irrational |
216 | $6\sqrt{6} \approx 14.697$ | Irrational |
294 | $7\sqrt{6} \approx 17.146$ | Irrational |
Every row with a $\sqrt{6}$ shares the same irrational part, only the whole-number multiplier changes.
Where the Square Root of 384 Appears
The square root of 384 shows up as the side length of a square whose area is $384$ square units: that side is $\sqrt{384} = 8\sqrt{6} \approx 19.596$ units. It also appears in geometry problems built on the number $6$ under a radical, since $384$ is $64 \times 6$, and $64$ is a clean perfect square.
Where the Square Root of 384 Appears
The square root of 384 shows up as the side length of a square whose area is $384$ square units: that side is $\sqrt{384} = 8\sqrt{6} \approx 19.596$ units. It also appears in geometry problems built on the number $6$ under a radical, since $384$ is $64 \times 6$, and $64$ is a clean perfect square.
What Is a Square Root?
The square root of a number $n$ is the value that, multiplied by itself, returns $n$. In symbols, $\sqrt{n} = x$ means $x^2 = n$.
A square root simplifies when the number hides a perfect-square factor. For 384, that hidden factor is $64$, and pulling it out is what turns $\sqrt{384}$ into $8\sqrt{6}$.
How to Find the Square Root of 384
Two methods matter here: prime factorisation to get the exact simplified form, and long division to get the decimal.
Method 1: Prime factorisation (simplify to exact form)
Break 384 into primes: $$384 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 = 2^7 \times 3$$
Group the twos into pairs: $$2^7 = (2^2)(2^2)(2^2)(2) = 64 \times 2$$
So $384 = 64 \times 6$.
Take the perfect square $64$ out of the radical: $$\sqrt{384} = \sqrt{64 \times 6} = 8\sqrt{6}$$
Final answer: $\sqrt{384} = 8\sqrt{6}$.
Method 2: Long-division method (estimate the decimal)
Find the perfect squares on either side of 384: $$19^2 = 361 \qquad 20^2 = 400$$
Since $361 < 384 < 400$, the answer lies between $19$ and $20$.
Test $19.6$: $19.6^2 = 384.16$.
That is just above 384, so the value is a touch under $19.6$.
Refining gives $\approx 19.596$.
Final answer: $\sqrt{384} \approx 19.596$.
Common Mistakes With Square Root of 384
Mistake 1: Leaving the radical only partly simplified
Where it slips in: spotting one small square factor and stopping. Don't do this: write $\sqrt{384} = 2\sqrt{96}$ and call it done. The correct way: keep factoring until nothing square remains. $\sqrt{96}$ still simplifies, so $2\sqrt{96} = 8\sqrt{6}$. The first instinct is to pull out the smallest square; the fully simplified form is $8\sqrt{6}$.
Mistake 2: Pairing the wrong number of factors
Where it slips in: counting the seven twos in $2^7$. Don't do this: treat $2^7$ as three pairs with none left, giving $8\sqrt{3}$. The correct way: three pairs use six twos and leave one two behind, which joins the $3$ to make $6$ under the radical, giving $8\sqrt{6}$.
Mistake 3: Rounding before simplifying
Where it slips in: reaching for a calculator first. Don't do this: record $19.596$ and lose the exact form. The correct way: find $8\sqrt{6}$ first, then round if a decimal is needed. The exact form stays precise through later steps.
Conclusion
The square root of 384 simplifies to $8\sqrt{6} \approx 19.596$.
It is irrational because $384 = 2^7 \times 3$ has an unpaired prime factor under the radical.
The trick is spotting $384 = 64 \times 6$, then taking $\sqrt{64} = 8$ out front.
Fully simplify before rounding, and count the pairs of twos carefully so you land on $8\sqrt{6}$, not $8\sqrt{3}$.
To take radicals further with a teacher, explore Bhanzu's algebra tutor or online math classes.
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