What is the Cube Root of a Number?
Before we learn what a cube root is, let’s first understand what a perfect cube is. A perfect cube is the product obtained when a number is multiplied by itself three times.
For instance, when a number x is multiplied by itself three times, we get the product as y. So we can say that y is a perfect cube of x. Using the same example, x is said to be the cube root of y. Hence, we can see that cubing and finding the cube root are inverse operations. Cube roots are denoted by the symbol “\(\sqrt[3]{}\)”. Hence, to symbolically denote that the cube root of y is x, we can write it as \(\sqrt[3]{y}=x\).
So, when considering 343 whose cube root is 7, we can write the cube root as: \(\sqrt[3]{343}=7\).
How to Find the Cube Root of 343
We can find the cube root of 343 by prime factorization.
A factor tree is used to find the prime factors of 343:
Therefore, 343 = 7 x 7 x 7 = \(7^3\)
So, \(\sqrt[3]{343}=\sqrt[3]{7^3}=7\)
Therefore, \(\sqrt[3]{343}=7\)
Solved Examples
Example 1: James had a cubical water tank, the amount of water that can be filled in the tank is 343 cubic inches. Find the length of the edge of the tank.
Solution:
The volume of the cubical tank = 343 cubic inches
The volume of a cube can be found using the formula,
Volume of cube, V = \(side^3\)
343 = \(side^3\)
Taking the cube root of each side,
\(\sqrt[3]{343}=\sqrt[3]{side^3}\)
7 = side
So, the edge length of the tank is 7 inches.
Example 2: Find the cube root of 512.
Solution:
Use a factor tree to find the prime factorization of 512:
Therefore, 512 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = \(2^9\)
So, \(\sqrt[3]{512}=\sqrt[3]{2^9}=2^3=8\)
Therefore, \(\sqrt[3]{512}=8\)
So, the cube root of 512 is 8.
Example 3: Find the value of \(2\sqrt[3]{343}~-~2\sqrt[3]{8}\).
Solution:
We know that, \(\sqrt[3]{343}=7\) and \(\sqrt[3]{8}=2\).
So, \(2\sqrt[3]{343}~-~2\sqrt[3]{8}\) = 2 x 7 – 2 x 2
= 14 – 4
= 10
So, \(2\sqrt[3]{343}~-~2\sqrt[3]{8}\) = 10.
Yes, 7 when multiplied by itself 3 times result in 343, 7 x 7 x 7 = 343 that is, \(\sqrt[3]{343}=7\). Therefore, 343 is a perfect cube.
The cube root of an odd number is always odd and the cube root of an even number is always even. So, the cube root of 343 will be an odd number, as 343 is odd. \(\sqrt[3]{343}=7\), so the cube root of 343 is 7, which is an odd number.
The value of the power when the exponent is \(\frac{1}{3}\) is the cube root of the base.