What Is A Cube Root?
The cube root of a number $n$ is the value $r$ that satisfies $r^3 = n$. Written $\sqrt[3]{n}$, it undoes the operation of cubing, just as a square root undoes squaring.
Every real number has exactly one real cube root, and the sign is kept: $\sqrt[3]{8} = 2$ while $\sqrt[3]{-8} = -2$. That single-value behaviour is different from square roots, which split into a positive and a negative branch.
Where Does The Cube Root Of 250 Appear?
$\sqrt[3]{250}$ shows up whenever a volume of 250 cubic units has to become a side length. A cube holding 250 cm³ of water has an edge of $5\sqrt[3]{2} \approx 6.30$ cm, since edge $= \sqrt[3]{\text{volume}}$.
It also appears in scaling problems: doubling a cube's volume from 125 to 250 multiplies each edge by $\sqrt[3]{2}$, the same irrational factor that runs through the classic "doubling the cube" problem from ancient Greek geometry.
Quick Reference Table
The cube root of 250 sits between two whole-number cube roots. Reading the neighbours makes the estimate feel concrete.
Number $n$ | $\sqrt[3]{n}$ (approx.) | Simplified radical |
|---|---|---|
27 | 3.0000 | $3$ (perfect cube) |
125 | 5.0000 | $5$ (perfect cube) |
250 | 6.2996 | $5\sqrt[3]{2}$ |
343 | 7.0000 | $7$ (perfect cube) |
512 | 8.0000 | $8$ (perfect cube) |
1000 | 10.0000 | $10$ (perfect cube) |
Is The Cube Root Of 250 Rational Or Irrational?
$\sqrt[3]{250}$ is irrational. It cannot be written as a fraction $\frac{p}{q}$ of two integers, and its decimal never terminates or repeats.
The reason is structural. A whole number has a rational cube root only when it is a perfect cube, meaning every prime in its factorization appears a multiple of three times.
The prime factorization of 250 is:
$$250 = 2 \times 5 \times 5 \times 5 = 2 \times 5^3.$$
The $5^3$ is a clean triple, but the lone $2$ is not. Because one prime is left with an exponent that is not a multiple of three, 250 is not a perfect cube, so its cube root carries the irrational factor $\sqrt[3]{2} \approx 1.2599$.
How Do You Find The Cube Root Of 250? (Prime Factorization And Estimation)
Two methods work together: prime factorization gives the exact simplified form, and estimation between perfect cubes gives the decimal.
Method 1: Prime factorization (exact form).
$$250 = 2 \times 5^3.$$
$$\sqrt[3]{250} = \sqrt[3]{5^3 \times 2}.$$
$$\sqrt[3]{250} = \sqrt[3]{5^3} \times \sqrt[3]{2}.$$
$$\sqrt[3]{250} = 5\sqrt[3]{2}.$$
Method 2: Estimation between perfect cubes (decimal value).
$$5^3 = 125 \quad \text{and} \quad 6^3 = 216 \quad \text{and} \quad 7^3 = 343.$$
$$216 < 250 < 343 \Rightarrow 6 < \sqrt[3]{250} < 7.$$
Since 250 sits closer to 216 than to 343, the root is just above 6.
$$6.3^3 = 250.047 \Rightarrow \sqrt[3]{250} \approx 6.2996.$$
The exact answer, $5\sqrt[3]{2}$, is what you keep in algebra; $6.2996$ is what you use when a decimal is required. Because $250^{1/3}$ is the same object written with a rational exponent, a calculator's $x^y$ key with $y = 1/3$ returns the same value.
Examples Of The Cube Root Of 250
The set builds from the plain simplification to a doubling-the-cube application, with one common wrong turn shown in full.
Example 1
Simplify $\sqrt[3]{250}$ into radical form.
$$250 = 2 \times 5^3.$$
$$\sqrt[3]{250} = 5\sqrt[3]{2}.$$
Final answer: $5\sqrt[3]{2}$.
Example 2
A student simplifies $\sqrt[3]{250}$ and writes $\sqrt[3]{250} = \sqrt[3]{2} \times \sqrt[3]{5}$. Is that the simplest form?
Wrong attempt. The student factors 250 as $2 \times 125$, then writes $\sqrt[3]{2} \times \sqrt[3]{125}$, but stops at $\sqrt[3]{2} \times \sqrt[3]{5}$ by mistake, splitting 125 as if it were 5.
Why it breaks. $\sqrt[3]{5}$ is not 5. Cubing the claimed answer gives $2 \times 5 = 10$, not 250, so the split is wrong.
Correct. The perfect cube inside 250 is $125 = 5^3$, and $\sqrt[3]{125} = 5$ exactly.
$$\sqrt[3]{250} = \sqrt[3]{2} \times \sqrt[3]{125} = 5\sqrt[3]{2}.$$
Final answer: $5\sqrt[3]{2}$.
Example 3
Estimate $\sqrt[3]{250}$ to one decimal place without a calculator.
$$6^3 = 216 \quad \text{and} \quad 6.3^3 = 250.047.$$
Since $250.047$ is barely above 250, the root is a shade under 6.3.
Final answer: $\sqrt[3]{250} \approx 6.3$.
Example 4
A cube-shaped tank holds 250 litres. What is its edge length, using $1 \text{ litre} = 1 \text{ dm}^3$?
$$\text{edge} = \sqrt[3]{250} \text{ dm}.$$
$$\text{edge} = 5\sqrt[3]{2} \approx 6.30 \text{ dm} = 63.0 \text{ cm}.$$
Final answer: about $63.0$ cm per edge.
Example 5
Evaluate $2 + 3\sqrt[3]{250}$ as a decimal to two places.
$$3\sqrt[3]{250} = 3 \times 6.2996 = 18.8988.$$
$$2 + 18.8988 = 20.8988.$$
Final answer: $\approx 20.90$.
A note from the classroom: the first instinct on $\sqrt[3]{250}$ is to reach straight for a calculator, but the more reliable move is to spot the largest perfect-cube factor first. Learners who write the prime factorization before pulling anything out almost never lose the leftover 2.
Common Mistakes
Mistake 1: Treating 250 as a perfect cube
Where it slips in: Reading "250" and assuming a whole-number answer exists.
Don't do this: Writing $\sqrt[3]{250} = 6$ because $6^3 = 216$ is "close enough."
The correct way: 250 is not a perfect cube, so the exact answer stays as $5\sqrt[3]{2}$, and the decimal is $6.2996$, not a whole number.
Mistake 2: Pulling out the wrong factor
Where it slips in: Simplifying without a full prime factorization.
Don't do this: Splitting $250 = 2 \times 125$ and then writing $\sqrt[3]{125} = 25$ or forgetting the leftover 2.
The correct way: Factor completely to $2 \times 5^3$. Only a full triple of the same prime leaves the radical, so $\sqrt[3]{5^3} = 5$ and the 2 stays inside. The habit that fixes this is writing the factorization first, every time.
Mistake 3: Confusing cube roots with square roots
Where it slips in: Copying the "two answers" rule from square roots.
Don't do this: Writing $\sqrt[3]{250} = \pm 5\sqrt[3]{2}$.
The correct way: A cube root has exactly one real value and keeps its sign. $\sqrt[3]{250}$ is positive; only $\sqrt[3]{-250} = -5\sqrt[3]{2}$ is negative. This is the same distinction that trips readers moving between $\sqrt[3]{n}$ and $\sqrt{n}$, so keep the index in view.
Conclusion
The cube root of 250 is $5\sqrt[3]{2}$, about $6.2996$, and it is irrational.
The simplification comes from $250 = 2 \times 5^3$: the perfect cube $5^3$ leaves the radical, the 2 stays.
Estimate the decimal by trapping 250 between $6^3 = 216$ and $7^3 = 343$.
The most common slip is treating 250 as a perfect cube or splitting 125 wrongly.
To work through radicals like this with a teacher, explore Bhanzu's algebra tutor, sessions with a high school math tutor, or live math classes online. Want a live Bhanzu trainer to walk through more cube-root problems? Book a free demo class.
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