Cube Root of 729 Explained

Introduction to Cube Root of 729

In terms of algebra and arithmetic, the cube of a given number ‘n’ is its 3rd power: that is, the result of the number is raised to power three by multiplying itself twice: n³ = n × n × n. It can also be mathematically formulated as the number multiplied by its square: n³ = n × n². It is also referred to as the volume formula for a geometric cube that has sides of length x, giving rise to the name. Furthermore, the inverse operation of identifying a number whose cube is x is called deriving the cube root of x. It identifies the side of the cube of a given volume, which is x raised to the 1/3rd power. Let’s simplify the explanation with an example of a cube root of 729. ∛729 = 9³ = 9 ×9 × 9 = 729.

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Notation of Cube Root

It is possible to write down the cube root in a different way, which also sometimes appears to be more convenient. Since a cube root is a unique case of an exponent. It can alternatively be expressed as follows:

∛ (n) =N Cube Root ^ (1/3)

Keep in mind that mostly the cube root will not be a rational number. Such numbers can be stated in the form of a quotient of two natural numbers, i.e., a fraction. Fractions at times can cause some complications, particularly when it comes to adding them. If you are having difficulty finding a common denominator of two fractions, check out the LCM method, which considers the least common multiple of given two numbers.

Perfect Cubes and Non-Perfect Cubes

It is straightforward to find the cube root of perfect cubes. However, most other numbers do not make for perfect cubes but are still used often. Thus, we will tell you the simplest way to find the rood of non-perfect and bigger cubes. For the same, below is a collection of perfect and non-perfect cubes rounded up to the hundredths:

List of Perfect Cubes

List of Non - Perfect Cubes

Simplification Process to Find Cube Roots Radical Form and Derivative

Follow the following step by step procedure to find the cube root of any given number easily. First, find all factors under the cube root. Let’s take an example for number 729 has the cube factor of 81. Let's take a look at this width ∛81×1= ∛729. As you can observe, the radicals are not in their purest form. Now derive and take off the cube root ∛81 × ∛1. Cube of ∛81=9, which results in 9 ∛. Now all radicals are simplified, meaning that they no longer have any cube factors.

Example of Irrational, Non-Perfect Cubes

Let’s say to find out the cube root of 30?

Given that the number is 30, which is not a perfect cube, so we begin finding the cube root of 30 the way below.

Firstly, factor the nearby numbers i.e.

We can do [3 × 3 × 3] = 27 and

[4 × 4 × 4] = 64,

That being so, we can conclude the answer (cube root of 30) will be between 3 and 4.

Let's try 3.1: (3.1 × 3.1 × 3.1) = 29.791

Let's try 3.2: (3.2 × 3.2 × 3.20 = 32.768

Let's try 3.3: (3.3 × 3.3 × 3.3) = 35.93

The nearest value we are getting is 3.1, that equals 29.79, and we want the cube root of 30, so we can also try doing with 3.11, which brings us 30.08. So our answer is only an approximation!

Note: These kinds of numbers are mathematically called surds, which are a peculiar type of irrational number.

Solved Examples of Cube Root

Example1:

Find the Cube Root of a Negative Perfect Cube of (-10648)

Solution1:

For a positive integer, we take an equivalent negative integer.

We know that ∛-10648 = -∛10648

Resolving 10648 into prime factors, we have

10648 = 2 × 2 × 2 × 11 ×11 × 11

= [2 × 2 × 2] × [11 × 11 × 11]

Therefore, ∛10648 = (2 × 11) = 22

Thus, ∛-10648 = - (∛10648) = -22

Example2:

Determine the Cube Root of Product of Integers ∛(729 × 125).

Solution2:

For the Cube Root of Product of Integers, We have (∛729 × ∛125).

= (∛729 × 125)

= ∛729 × ∛125

= [∛ {9 × 9 × 9}] × [∛{5 × 5 × 5}]

= (9 × 5)

= 45

Thus Cube Root of Product of Integers (∛729 × ∛125) = 45.