264 LearnersLast updated on May 26th, 2025
Cube Root of 66
We will learn the cube root concept to use it on other mathematical topics like algebra, mensuration, geometry, trigonometry, etc. So, it is as important as learning square roots. Let us now see how we can obtain the cube root value of 66, and its examples.
What Is the Cube Root of 66?
The cube root of 66 is the value which, when multiplied by itself three times (cubed), gives the original number 66. The cube root of 66 is 4.04124002062. The cube root of, 66 is expressed as β66 in radical form, where the “ β ” sign” is called the “radical” sign. In exponential form, it is written as (66)β
. If “m” is the cube root of 66, then, m3=66. Let us find the value of “m”.
Finding the Cubic Root of 66
We can find cube roots of 66 through a method, named as, Halley’s Method. Let us see how it finds the result.
Cube Root of 66 By Halleyβs 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 βa≅ 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, 66.
Step 1: Let a=66. Let us take x as 4, since 43=64 is the nearest perfect cube which is less than 64.
Step 2: Apply the formula. β64≅ 4((43+2×66) / (2(4)3+66)) = 4.04…
Hence, 4.04… is the approximate cubic root of 66.
Common Mistakes and How to Avoid Them in the Cube Root of 66
Understanding common misconceptions or mistakes can make your calculations error free. So let us see how to avoid those from happening.
Mistake 1
For β66, students can make error in the important part of prime factorization.
Prime Factorization is a crucial part in determining cube roots and square roots. So, use the exact prime factors and group (or pair).
Mistake 2
Applying exponent rules incorrectly when finding cube roots.
β66 is not same as 663. β66 = 4.04… and, 663 is 287496.
Mistake 3
For large non-perfect cubes, students frequently estimate wrongly and leave it with a wrong result.
Keep the methods of finding cube roots in front whenever attempting large non-perfect cubes.
Mistake 4
Students typically rely on memorizing rather than understanding concepts of cube root.
Memorizing is not the solution when it comes to complex problems. So understanding the methods of finding cube roots is advisable.
Mistake 5
Misapply the cube root property like when asked to solve β(66)3, students might do as β(66×3)
Clear the concepts again and again if they don’t understand, where to do what. β(66)3≠β(66×3). β(66)3= 66
Cube Root of 66 Examples
Problem 1
Find (β22/ β66) Γ (β22/ β66) Γ (β22/ β66)
(β22/ β66) × (β22/ β66) × (β22/ β66)
= (β22× β22× β22) / (β66× β66× β66)
=((22)β
)3/ ((66)β
)3
=22/66
=1/3
Answer: 1/3
Explanation
We solved and simplified the exponent part first using the fact that, β22=(22)β and β66=(66)β , then solved.
Problem 2
If y = β66, find yΒ³/ yβΉ
y=β66
⇒ y3/y9= (β66)3 / (β66)9
⇒ y3/y9= 66/ (66)3= 1/(66)2
Answer: 1/(66)2
Explanation
(β66)3=(661/3)3=66, and β(66)9=(661/3)9=(66)3. Using this, we found the value of y3/y9.
Problem 3
Multiply β66 Γ β1000
β66×β1000
= 4.04×10
=40.4
Answer: 40.4
Explanation
We know that the cubic root of 1000 is 10, hence multiplying β1000 with β66.
Problem 4
Solve the equation: (x)Β³/Β²=66
(x)3/2=66
⇒ (x)1/2=(66)1/3
⇒ (x)1/2 = 4.04
⇒ (x) = (4.04)2
⇒ x= 16.3216
Answer: 16.3216
Explanation
Simplified the equation applying the cube root features and found the answer.
FAQs on 66 Cube Root
1.What is the nearest cube to 66?
2.What is the square root of 66?
3.What is β66 simplified ?
4.How to simplify the cube root of 66?
5.Is 66 a real number?
Important Glossaries for Cube Root of 66
- Cube root properties - The features when cube root is applied to any number. Those are: 1) The cube root of all odd numbers is an odd number. The same applies for even numbers also, that is, the cube of any even number is even.
2) The cube root of a negative number is also negative.
3) If the cube root of a number is a whole number, then that original number is said to be perfect cube
- Irrational Numbers - Numbers which cannot be expressed as m/n form, where m and n are integers and n not equal to 0, are called Irrational numbers.
- Square root -The square root of a number is a number which when multiplied by itself produces the original number, whose square root is to be found out.
- 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 near to the correct answer, but not perfectly correct.
- Iterative method - This method is a mathematical process which uses an initial value to generate a further sequence of solutions for a problem, step-by-step.
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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.
