Cube Root of 75
2026-02-28 09:06 Diff
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Last updated on August 5, 2025

The cube root of 75 is the value that, when multiplied by itself three times (cubed), gives the original number 75. Do you know? Cube roots apply to our real life also, like that for measuring dimensions, density and mass, field of engineering etc.

What Is the Cube Root of 75 ?

The cube root of 75 is 4.21716332651. The cube root of 75 is expressed as ∛75 in radical form, where the “∛"  sign is called the “radical” sign. In exponential form, it is written as (75)⅓. If “m” is the cube root of 75, then, m3=75. Let us find the value of “m”.

Finding the Cube Root of 75

The cube root of 75 is expressed as ∛75 as its simplest radical form,

since 75 = 5×5×3


∛75 = ∛(5×5×3)


Group together three same factors at a time and put the remaining factor under the ∛ .


∛75= ∛75 


We can find cube root of 75 through a method, named as, Halley’s Method. Let us see how it finds the result.
 

Cube Root of 75 By Halley’s Method

Now, what is Halley’s Method? It is an iterative method for finding cube roots of a given number N, such that, x3=N,

where this method approximates the value of “x”.


Formula is ∛a≅ x((x3+2a) / (2x3+a)), where 


a=given number whose cube root you are going to find


x=integer guess for the cubic root


 Let us apply Halley’s method on the given number 75.


Step 1: Let a=75. Let us take x as 4, since, 43=64 is the nearest perfect cube which is less than 75.


Step 2: Apply the formula.  ∛75≅ 4((43+2×75) / (2(4)3+75))= 4.12…


Hence, 4.12… is the approximate cubic root of 75.
 

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Common Mistakes and How to Avoid Them in the Cube Root of 75

Here are some common mistakes with their solutions given:
 

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Problem 1

Find (∛75/ ∛64) × (∛75/ ∛64) × (∛75/ ∛64)

Okay, lets begin

  (∛75/ ∛64) × (∛75/ ∛64) × (∛75/ ∛64)


= (∛75× ∛75× ∛75) / (∛64× ∛64× ∛64)


=((75)⅓)3/ ((64)⅓)3


=75/64

Answer: 75/64
 

Explanation

We solved and simplified the exponent part first using the fact that, ∛75=(75)⅓ and ∛64=(64)⅓ , then solved.

Well explained 👍

Problem 2

If y = ∛75, find y³.

Okay, lets begin

 y=∛75


⇒ y3= (∛75)3 


⇒ y3= 75


Answer: 75
 

Explanation

(∛75)3=(751/3)3=75. Using this, we found the value of y3.

Well explained 👍

Problem 3

Subtract ∛75 - ∛64

Okay, lets begin

∛75-∛64

= 4.217–4

= 0.217


Answer:     0.217
 

Explanation

We know that the cubic root of 64 is 4, hence subtracting  ∛64 from ∛75.
 

Well explained 👍

Problem 4

What is ∛(75⁶) ?

Okay, lets begin

 ∛(756)

= ((75)6))1/3

=( 75)2

= 5625


Answer: 5625 
 

Explanation

We solved and simplified the exponent part first using the fact that, ∛75=(75)⅓, then solved.
 

Well explained 👍

Problem 5

Find ∛(75+(-11)).

Okay, lets begin

 ∛(75-11)

= ∛64

= 4


Answer:     4
 

Explanation

Simplified the expression, and found out the cubic root of the result. 
 

Well explained 👍

FAQs on 75 Cube Root

1.∛75 lies between which two perfect cubes?

 ∛75=4.217… lies between perfect cubes 1 and 8.

2.Is 75 a perfect cube?

75 is not perfect cube since, ∛75=4.217… , 4.127… is not a whole number. 
 

3.What is the simplified form of ∛75?

∛75 = ∛(5×5×3) is the simplified form of ∛75.
 

4.What is the square root of 75?

The square root of 75 is ±8.6602… .
 

5.How to find a cube root without using a calculator?

We can find the cube root of a number by using methods like Halley’s Method or maybe prime factorization.
 

Important Glossaries for Cube Root of 75

  • Integers: Integers can be a positive natural number, negative of a positive number, or zero. We can perform all the arithmetic operations on integers. The examples of integers are, 1, 2, 5,8, -8, -12, etc.
  • Whole numbers: The whole numbers are part of the number system, which includes all the positive integers from 0 to infinity. 
  • Square root: The square root of a number is a value “y” such that when “y” is multiplied by itself → y ⤫ y, the result is the original number.
  • Polynomial: It is an algebraic expression made up of variables like “x” and constants, combined using addition, subtraction, multiplication, or division, where the variables are raised to whole number exponents.
  • Approximation: Finding out a value which is nearly correct, but not perfectly correct.
  • Iterative method: This method is a mathematical process which uses an initial value to generate further and step-by-step sequence of solutions for a problem.
     

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Jaskaran Singh Saluja

About the Author

Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.

Fun Fact

: He loves to play the quiz with kids through algebra to make kids love it.