Table Of Contents
Cube Root Definition
As stated, the cube root of a number is a value that, when multiplied by itself three times, results in the original number. So, if x is the cube root of y, then we can write,
\(x \times x \times x = y.\)
Cube root of a number is represented by the symbol \(‘\sqrt[3]{ }’.\) So for the equation above, we can write,
\(\sqrt[3]{y } = x\)(read as, the cube root of y is x).
Additionally, the exponential representation of cube roots is, \(y^{1/3} = x.\)
Cube Root of 3
The exact value of the cube root of 3, that is, \(\sqrt[3]{3}\text{ }or\text{ }3^{1/3}\)is 1.44224957…. The value is a non-terminating and a non-repeating decimal. We can follow the given steps to arrive at the approximate value of cube root of 3. This method is known as the approximation method.
Step 1: Assume that the cube root of 3 is x. So,
\(\sqrt[3]{3}\text{ }= x.\)
Step 2: We know, \(1^{3} = 1\text{ }and\text{ }2^{3}\text{ }= 8,\)so the cube root of 3, that is, x will lie between 1 and 2. Also, it will be closer to 1 than to 2 on the number line. Assume a value for x, say, x is approximately equal to 1.4.
Step 3: Divide 3 by 1.4. We get,
\(3 \div 1. 4 = 2.1428.\)
Step 4: Again divide the quotient obtained in step 3 by 1.4. We get,
\(2.1428 \div 1. 4 = 1.5306.\)
Step 5: Calculate the average of 1.4, 1.4 and 1.5306.
\(\frac{1.4 + 1.4 + 1.5306}{3} = 1.44\)
So, the approximate value of the cube root of 3 is 1.44 (≈1.4424957…).
Rapid Recall
Solved Examples
Example 1: Find the side length of a cube with volume 3 cubic inches.
Solution:
Let the side length of the cube be x inches.
Volume of a cube \(= x^{3}\) [Volume of cube formula]
\( x^{3} = 3\) [Substitute 3 for volume]
\(\Rightarrow x = \sqrt[3]{3}\) [Cube root on both sides]
\(\Rightarrow x \approx 1.44\text{ }in.\) [Approximate value of cube root of 3]
So, the side length of the cube is about 1.44 inches.
Example 2: The volume of a sphere is 12.56 cubic centimeters.Find the radius of the sphere. Take \(\pi\) as 3.14.
Solution:
Volume of a sphere \(= \frac{4}{3} \pi r^{3}\) [Volume of sphere formula]
\(= 12.56 =\frac{4}{3} \pi r^{3}\) [Substitute 12.56 for volume]
\(\Rightarrow r^{3} = \frac{3}{4\pi }.12.56\) [Solve for radius, r]
\(\Rightarrow r^{3} = \frac{3}{4 \times 3.14 }.12.56\) [Substitute 3.14 for \(\pi \)]
\(\Rightarrow r^{3} = 3\) [Simplify]
\(\Rightarrow r \approx 1.44\text{ }cm\) [Cube root on both sides]
So, the radius of the sphere is about 1.44 centimeters.
The cube root of 3 is not an integer. Hence 3 is not a perfect cube.
If the volume of a cube or a sphere is given then we can use the cube roots of numbers to find the edge length of the cube or the radius of the sphere.
The cube root of 3 cannot be expressed in a/b form where a,b are integers and b≠0. So the cube root of 3 is not a rational number.
In radical form the cube root of 3 is written as ∛3.
The cube root of a positive number is always positive and that of a negative number is negative. This is in line with the fact that the sign of the product obtained by multiplying a number three times by itself is the same as the sign of the number.