Introduction to Cube Roots
Cube of a number
The result of multiplying an integer by itself three times is known as its cube. In fact, the volume of the geometric figure cube is the cube of the measure of its side.
To define it simply, a cube number is a number with the exponential power of three. Since in determining the cube of a number the number is multiplied with itself 3 times, the cube of a positive number is positive and the cube of a negative number is negative.
If the number is b, then its cube is given by the equation,
\(b\times\ b\times\ b=b^3\) or
\(b^3\ =\ a\) (writing in exponential form, where the number b is called the base, 3 is the exponent and a is the cube of b)
For example, the cube of 7 is \(\ 7\times7\times7\ =\ 7^3\) (writing in exponential form)
\(7^3 = 343\)
So, \(\sqrt[3]{343}=7\) (solved) \((\sqrt[3]{a}=b)\)
For example, the cube of -5 is \(-5\times\ -5\times\ -5\ =\ -5^3\) (writing in exponential form)
\(( -b\times\ -b\times\ -b={-b}^3)\)
\(-5^3 =-125\)
What is a Cube root?
Definition: The cube root of any number ‘a’ is a number ‘b’ whose cube is equal to ‘a’. The factor that we multiply by itself three times to get a number is called the cube root. Cube root is represented with the symbol ‘\(\sqrt[3]{}\)’. This can be represented as an equation, \(\sqrt[3]{a}=b\) (that is, the cube root of a number ‘a’ is ‘b’)
If ‘a’ is negative, then
\(\sqrt[3]{-a}=\ -b\)
For example ,
\(\sqrt[3]{343}=7\) (taking \(7^3 = 343\)) and
\(\sqrt[3]{-125}=-5\) (taking \(-5^3 =-125\))
Hence, we can say that determining the cube and cube root are inverse operations.
That is,
\(b^3=a\) (cube)
\(\sqrt[3]{a}=b\) (cube root)
What is a Perfect cube?
A perfect cube is the value of a number that is equal to the cube of any integer.
So in equation \(\sqrt[3]{a}=b\), a is a perfect cube (where b is an integer).
Let us take an example: \(216\ =\ 6\times\ 6\ \times\ 6\ =\ 6^3\), so 216 is a perfect cube as it is equal to the cube of the number 6. However, for the number 210, there is no integer ‘n’ such that \(n^3=210\), so ‘210’ is not a perfect cube.
Below are given the perfect cube numbers between 1 to 1000.
\(1\ =\ 1\ \times1\ \times\ 1\) or \(1^3\)
\(8\ =\ 2\ \times\ 2\ \times\ 2\) or \(2^3\)
\(27\ =\ 3\ \times\ 3\ \times\ 3\) or \( 3^3\)
\(64\ =\ 4\ \times4\ \times\ 4\) or \( 4^3\)
\(125\ =\ 5\ \times\ 5\ \times\ 5\) or \( 5^3\)
\(216\ =\ 6\ \times\ 6\ \times\ 6\) or \(6^3\)
\(343\ =\ 7\times\ 7\ \times\ 7\) or \( 7^3\)
\(512\ =\ 8\ \times\ 8\ \times\ 8\) or \( 8^3\)
\(729\ =\ 9\ \times\ 9\ \times\ 9\) or \( 9^3\)
\(1,000\ =\ 10\ \times\ 10\ \times\ 10\) or \( {10}^3\)
We know that the measure of the side length of a square is the square root of its area. Similarly, the measure of the side length of a cube is the cube root of its area. We can use the inverse operation, that is, the cube of the length of the side to determine the volume of the cube.
For example, if we knew the volume of a cube was \({64\text{cm}}^3\), we’d know that each side would be 4 cm (since \(64\ =\ 4\ \times\ 4\ \times\ 4\) or \(4^3\)).
Similarly, a cube with a side length of 3 cm, would have a volume of \(27\text{cm}^3\) (because \(27\ =\ 3\ \times\ 3\ \times\ 3\) or \(3^3\)).
How to solve the equations using Cube roots?
Let’s understand how to solve equations having cube roots with the example below.
Solving equation
\(x^3=-729\)
\(x^3=-729\) (Writing the equation.)
\(\sqrt[3]{x^3}=\sqrt[3]{-729}\) (Taking the cube root of each side.)
\(x=-9\) (Simplified)
Hence, the result is \(x=-9\).
Solved Cube Root Examples
Example 1: Finding the cube root of \(\frac{1}{343}\).
Solution: \(\sqrt[3]{\frac{1}{343}}\)
As we know (\({\frac{1}{7})}^3=\frac{1}{343}\),
\(\sqrt[3]{\frac{1}{343}}=\frac{1}{7}\) (Simplified).
Example 2: Finding the cube root of 1728.
Solution: \(\sqrt[3]{1728}\)
as we know (\({12)}^3=1728\),
\(\sqrt[3]{1728}=12\) (Simplified).
Example 3: Solve the equation \({-\frac{1}{8}x}^3=8\)
Solution:
\({-\frac{1}{8}x}^3=8\) (Writing the equation.)
\(x^3=-64\) (multiplying each side by ‘-8’)
\(\sqrt[3]{x^3}=\ \sqrt[3]{-64}\) (Taking the cube root of each side.)
\(x=-4\) (Simplified since \(64\ =\ 4\ \times4\ \times\ 4\) or \(4^3\))
Hence, the result is \(x=-4\).
Example 4: A tennis ball is placed in a cube-shaped box. Calculate the volume of the box if it has sides of 6 centimeters. Also, calculate the surface area of the box.
Solution: The box is in the shape of a cube. Using the formula for volume (V) of a cube to find the volume of the cube given side l is 6cm.
\(V=l^3\) (writing the formula of the volume)
\(V=6^3\) (Substituting 6 in the place of l)
\(\sqrt[3]{V}=\sqrt[3]{6^3}\) (Taking the cube root on each side)
\(V=216~\text{cm}^3\) (simplified)
The side length is 6 centimeters. Using a formula for finding the surface area S of the cube.
\(S=6l^2\) (writing the formula for surface area)
\(S=6{(6)}^2\) (substituting 6 in the place of l)
\(S=216\) (simplified)
Hence, the volume of the box is 216 cm cubed and the surface area of the box is 216 sq. cm
Example 5: A cubic brick has a side length of 9 centimeters. Find the volume of the brick. Also, find its surface area.
Solution:
Using the formula for the volume of a cube to find the volume of the brick (V). We know the side length l is 9 cm.
\(V=l^3\) (writing the formula of the volume)
\(V=9^3\) (Substituting 9 in the place of l)
\(\sqrt[3]{V}=\sqrt[3]{9^3}\) (Taking the cube root on each side)
\(V=729\text{cm}^3\) (simplified)
Using the formula for finding the surface area S of the cube.
\(S=6l^2\) (writing the formula for surface area)
\(S=6{(9)}^2\) (substituting 9 in the place of l)
\(S=486\) (simplified)
Hence, the volume of the brick is \(729\text{cm}^3\) and the surface area of the cubic brick is 486 sq. cm
Prime factorisation method can be used to determine whether a given number is a perfect cube or not.
The cube root of the number 1 is the number itself, which is 1.