The cube root of 27 is 3 because multiplying 3 by itself three times gives 27:
[ 3 \times 3 \times 3 = 27 ]
In radical notation, this is written as:
[ \sqrt[3]{27}=3 ]
The small number 3 outside and above the radical sign is the index. Think about it: it tells us that the desired number must be used as a factor three times. Although people sometimes say “cubed root,” the standard mathematical term is cube root.
Introduction
Finding the cube root of 27 is a straightforward example of an inverse operation. But cubing a number means raising it to the third power, while taking a cube root reverses that process. Since (3^3=27), the real cube root is exactly 3.
The official docs gloss over this. That's a mistake.
This result is also easy to understand through geometry. And if a cube has a volume of 27 cubic units, each of its equal edges must measure 3 units. That relationship between volume and edge length makes the cube root useful in geometry, physics, engineering, and everyday measurement.
What Does “Cube Root” Mean?
A cube root of a number (a) is a value (x) that satisfies:
[ x^3=a ]
For 27, the question becomes:
[ x^3=27 ]
Testing 3 gives:
[ 3^3=3 \times 3 \times 3=27 ]
That's why, the principal real cube root of 27 is 3. The word principal matters because some equations have more than one type of solution. In ordinary arithmetic and real-number mathematics, however, (\sqrt[3]{27}) means the real answer, which is 3.
Unlike square roots, cube roots can be taken of negative numbers. Take this: (\sqrt[3]{-27}=-3), because:
[ (-3)(-3)(-3)=-27 ]
This works because three negative factors produce a negative result.
Step-by-Step Calculation
1. Translate the expression into an equation
Let the unknown cube root be (x):
[ x=\sqrt[3]{27} ]
This is equivalent to:
[ x^3=27 ]
2. Factor 27
Break 27 into prime factors:
[ 27=3 \times 9 ]
Since (9=3 \times 3), we have:
[ 27=3 \times 3 \times 3=3^3 ]
3. Apply the cube root
Substitute (3^3) for 27:
[ \sqrt[3]{27}=\sqrt[3]{3^3} ]
A cube root and a third power cancel for real numbers:
[ \sqrt[3]{3^3}=3 ]
Thus:
[ \boxed{\sqrt[3]{27}=3} ]
4. Verify the answer
Always check an inverse operation by reversing it:
[ 3^3=3 \times 3 \times 3=9 \times 3=27 ]
The verification returns the original number, confirming that the answer is correct.
Why the Answer Is Exactly 3
The number 27 is a perfect cube, meaning it is the cube of an integer. A few positive perfect cubes are:
- (1^3=1)
- (2^3=8)
- (3^3=27)
- (4^3=64)
- (5^3=125)
Because 27 appears in this pattern as (3^3), its cube root is the whole number 3. Not every number is a perfect cube. To give you an idea, (\sqrt[3]{30}) is slightly greater than 3, but it is not an integer.
Mathematical and Scientific Explanation
Exponent rules provide another way to express the calculation. Taking a cube root is equivalent to raising a number to the power of (\frac{1}{3}):
[ \sqrt[3]{27}=27^{1/3} ]
Since (27=3^3):
[ 27^{1/3}=(3^3)^{1/3} ]
When raising a power to another power, multiply the exponents:
[ (3^3)^{1/3}=3^{3 \times \frac{1}{3}}=3^1=3 ]
In science and engineering, cube roots often appear whenever a quantity depends on volume. The volume (V) of a cube with side length (s) is:
[ V=s^3 ]
To recover the side length from the volume, use the cube root:
[ s=\sqrt[3]{V} ]
If (V=27\text{ cm}^3), then:
[ s=\sqrt[3]{27\text{ cm}^3}=3\text
