What Is A Cube Root?
The cube root of a number $n$ is the value $r$ for which $r^3 = n$, the number that, multiplied by itself three times, returns $n$. So $\sqrt[3]{1331}$ is the number whose cube is 1331.
Here a whole number fits exactly: $11^3 = 11 \times 11 \times 11 = 1331$. The small index $3$ in $\sqrt[3]{\phantom{n}}$ is what separates a cube root from a plain square root.
Where Does ∛1331 Appear?
$\sqrt[3]{1331}$ is the natural answer to a volume question: a cube that holds 1331 cubic units has an edge length of exactly $\sqrt[3]{1331} = 11$ units. It also shows up in scaling, where enlarging a shape by a factor of 11 multiplies its volume by $11^3 = 1331$, so recovering the scale factor from a volume of 1331 is precisely this cube root.
Quick Reference Table
Number $n$ | $\sqrt[3]{n}$ | Perfect cube? |
|---|---|---|
729 | 9 | Yes ($9^3$) |
1000 | 10 | Yes ($10^3$) |
1331 | 11 | Yes ($11^3$) |
1728 | 12 | Yes ($12^3$) |
2197 | 13 | Yes ($13^3$) |
2744 | 14 | Yes ($14^3$) |
Is 1331 A Perfect Cube?
Yes. A perfect cube is an integer multiplied by itself three times, $1, 8, 27, 64, 125, \ldots, 1000, 1331$, and 1331 is on that list as $11^3$. You can read more about these values in cube numbers.
Because 1331 is a perfect cube, its cube root is a whole number, not an endless decimal. That is what makes $\sqrt[3]{1331}$ different from a root like $\sqrt[3]{1330}$, which is irrational.
Is The Cube Root Of 1331 Rational Or Irrational?
$\sqrt[3]{1331}$ is rational. It equals the whole number 11, which can be written as the fraction $\frac{11}{1}$, and every whole number is rational.
A number has a rational cube root exactly when it is a perfect cube. Since $1331 = 11^3$, its cube root closes cleanly to an integer, with no non-terminating tail, which matches the formal definition of a cube root.
How Do You Find The Cube Root Of 1331 By Prime Factorization?
For a cube root you group identical factors in threes; each complete triple sends one copy outside the radical. Here it is, one step per line.
$1331 = 11 \times 11 \times 11$
$1331 = 11^3$
$\sqrt[3]{1331} = \sqrt[3]{11^3}$
$\sqrt[3]{1331} = 11$
The three 11s form a single triple, and that triple leaves the radical as one 11, with nothing left inside. When every prime forms a complete group of three, the cube root is a whole number.
How Do You Find The Cube Root Of 1331 By The Unit-Digit Trick?
For a known perfect cube you can find the root without full factorization, a shortcut students on r/learnmath and Quora often ask about. It rests on how last digits behave when you cube a number.
Look at the unit digit of 1331, which is 1. A number ending in 1 cubes to a number ending in 1, so the root also ends in 1.
Strike out the last three digits of 1331, leaving the front group 1. Find the largest cube not exceeding that group: $1^3 = 1$, so the tens digit is 1.
Put the tens digit 1 with the unit digit 1 to get $\sqrt[3]{1331} = 11$. The same trick reads the cube root of 343 as 7, since 343 ends in 3, whose cube-root digit is 7.
Examples Of The Cube Root Of 1331
Example 1
Find $\sqrt[3]{1331}$ by prime factorization.
$1331 = 11^3$
$\sqrt[3]{1331} = 11$
Final answer: $11$
Example 2
A student guesses $\sqrt[3]{1331} = 111$ because the digits repeat, or reaches for a big number since 1331 looks large. Where does it go wrong?
The digit pattern of 1331 is a coincidence, not the root, and a large-looking number can still have a small root. Test the guess: $111^3$ is over a million, nowhere near 1331.
Anchor the estimate with nearby cubes instead: $10^3 = 1000$ and $12^3 = 1728$. Since $1331$ sits between them, the root is 11, confirmed by $11^3 = 1331$.
Final answer: $11$
Example 3
Evaluate $(\sqrt[3]{1331})^3$.
$(\sqrt[3]{1331})^3 = 1331$
Final answer: $1331$
Example 4
A cubical tank has a volume of 1331 cubic feet. Find its edge length.
edge $= \sqrt[3]{1331}$
edge $= 11$
Final answer: $11$ feet.
Example 5
Simplify $\sqrt[3]{1331} + \sqrt[3]{1000}$.
$\sqrt[3]{1331} + \sqrt[3]{1000} = 11 + 10$
$\sqrt[3]{1331} + \sqrt[3]{1000} = 21$
Final answer: $21$. Both are perfect cubes, so each root is a whole number and the sum is a clean integer.
Common Mistakes
Mistake 1: Confusing the cube root with the square root
Where it slips in: Ignoring the small index 3 in the radical.
Don't do this: Reading $\sqrt[3]{1331}$ as $\sqrt{1331} \approx 36.48$.
The correct way: The index 3 asks "what cubes to 1331," which is 11. Always check the index before you compute.
Mistake 2: Pulling out pairs instead of triples
Where it slips in: Treating a cube root like a square root and grouping in twos.
Don't do this: Writing $\sqrt[3]{1331} = 11\sqrt[3]{11}$ by taking out one pair of 11s.
The correct way: A cube root releases a factor only for a complete group of three. The three 11s form one triple, so a single 11 comes out with nothing left inside, giving exactly 11.
Mistake 3: Calling the answer irrational
Where it slips in: Assuming every root is a messy decimal.
Don't do this: Reporting $\sqrt[3]{1331} \approx 11.0$ as an approximation.
The correct way: 1331 is a perfect cube, so $\sqrt[3]{1331}$ is exactly 11, a rational whole number, not a rounded value.
Conclusion
The cube root of 1331 is exactly 11, a rational number, because $1331 = 11^3$ is a perfect cube. Group the primes in threes to confirm it, or use the unit-digit trick to read the root off in seconds. To build these radical and exponent skills with a teacher, explore Bhanzu's algebra tutor, work with a middle school math tutor, or join live math classes online.
Want to practice with a guide? Book a free demo class and work through cube roots step by step.
Read More
Cube Root 1 to 100, the full reference table of cube roots from 1 to 100.
Cube Root of 27, a clean perfect cube, exactly 3.
Cube Root of 64, another perfect cube, exactly 4.
Cube Root of 100, a nearby non-perfect cube that is irrational, for contrast.
Squares and Square Roots, how roots and powers connect across indices.
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