Last updated on May 26th, 2025
The cube root of 192 is the value that, when multiplied by itself three times (cubed), gives the original number 192. Do you know? Cube roots apply to our real life also, like that for measuring volume and scaling, density and mass calculation, etc.
The cube root of 192 is 5.76899828123. The cube root of 192 is expressed as β192 in radical form, where the “β" sign is called the “radical” sign. In exponential form, it is written as (192)β . If “m” is the cube root of 192, then, m3=192. Let us find the value of “m”.
The cube root of 192 is expressed as 4β3 as its simplest radical form, since
192 = 2×2×2×2×2×2×3
β192 = β(2×2×2×2×2×2×3)
Group together three same factors at a time and put the remaining factor under the β .
β192= 4β3
We can find cube root of 192 through a method, named as, Halley’s Method. Let us see how it finds the result.
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 192.
Step 1: Let a=192. Let us take x as 5, since, 125 is the nearest perfect cube which is less than 192.
Step 2: Apply the formula. β192≅ 5((53+2×192) / (2(5)3+192))=5.757
Hence, 5.757 is the approximate cubic root of 192.
some common mistakes with their solutions are given below:
Find β192/ β8
β192/ β8
= 5.768 / 2
= 2.884
Answer: 2.884
We know that the cubic root of 8 is 2, hence dividing β192 by 2.
The Volume of a cube is 192 cubic centimeters, find the length of one side of the cube.
We know that, (side of a cube)3=Volume of a cube
⇒side of the cube = β(Volume of the cube)
⇒side of the cube = β192
⇒ side of the cube = 5.768 cm
Answer: 5.768 cm
We applied the formula for finding the volume of a cube, and inverted it to find the measure of one side of the cube.
Subtract β192 - β125
β192-β125
= 5.768–5
= 0.768.
Answer: 0.768
We know that the cubic root of 8 is 2, hence subtracting β125 from β192.
What is β(192Β²) ?
β(1922)
= β36864
= 33.281…
Answer: 33.281…
We first found the square value of 192, which is 36864, and then found out the cube root of 36864.
Find β(192+24).
β(192+24)
= β216
=6
Answer: 6
Simplified the expression, and found out the cubic root of the result.
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.
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