Cube Root of 748

We have learned the definition of the cube root. Now, let’s learn how it is represented using a symbol and exponent. The symbol we use to express the cube root is the radical sign (∛), and the exponent we use is ⅓.

In exponential form, ∛748 is written as 748^(1/3). The cube root is just the opposite operation of finding the cube of a number. For example, assume ‘y’ is the cube root of 748, then y³ can be 748. Since the cube root of 748 is not an exact value, we can write it as approximately 9.0807.

Finding the cube root of a number involves identifying the number that must be multiplied three times to result in the target number. Now, we will go through different ways to find the cube root of 748. The common methods we use to find the cube root are given below:

• Prime factorization method
• Approximation method
• Subtraction method
• Halley’s method

To find the cube root of a non-perfect number, we often follow Halley’s method. Since 748 is not a perfect cube, we use Halley’s method.

Let's find the cube root of 748 using Halley’s method. The formula is: ∛a ≅ x((x³ + 2a) / (2x³ + a))

where: a = the number for which the cube root is being calculated

x = the nearest perfect cube

Substituting,

a = 748;

x = 9

∛a ≅ 9((9³ + 2 × 748) / (2 × 9³ + 748))

∛748 ≅ 9((729 + 2 × 748) / (2 × 729 + 748))

∛748 ≅ 9.0807

The cube root of 748 is approximately 9.0807.

• Cube root: The number that is multiplied three times by itself to get the given number is the cube root of that number.
   
• Perfect cube: A number is a perfect cube when it is the product of multiplying a number three times by itself. A perfect cube always results in a whole number. For example, 3 × 3 × 3 = 27; therefore, 27 is a perfect cube.
   
• Exponent: The exponent form of the number denotes the number of times a number can be multiplied by itself. In a^(1/3), ⅓ is the exponent which denotes the cube root of a.
   
• Radical sign: The symbol that is used to represent a root, expressed as (∛).
   
• Irrational number: Numbers that cannot be expressed as a simple fraction are irrational. For example, the cube root of 748 is irrational because its decimal form goes on continuously without repeating numbers.