Cubes and Cube Roots are commonly used in solving mathematical problems. Remembering and applying these concepts of Cubes and Cube Roots quickly in problem-solving techniques give trade-offs in competitive exams.

When a positive number is multiplied three times by itself then the result is said to be Cube of that number, on the other hand, if Cube Root of the result and it will result in the same positive number. For Example, if x is the cube root of y, then it’s shown as x = ∛y or it can be shown as x³ = y.

Numeric Example: If 64 is the cube root of 4, then it can be shown as 4 = ∛64 or it can be shown as 4³ = 64. It’s easier to find out the cube of the given number but it requires effort and the right method to find out the Cubes and Cube Roots of any given number which is not a cube of a perfect number.

Cubes

The cube of a number is the product of the multiplication of the number thrice. For Example: If a number is expressed as ‘x’ then the cube of ‘x’ is shown as x multiplied thrice by x and the product is x³. Cube Numbers are also called Perfect Cube Numbers.

The product of numbers which is not the same is not called a perfect cube number. For example: 5 x 6 x 10 = 300, 300 is not a perfect number. Whereas, 5 x 5 x 5 = 125, hence 125 is called a perfect cube number.

Another easy way to find the cube of a number is to multiply the square of that number by that number. For example, if we have to find a cube of the number 22, then if we remember the square of 22 that is 484, then multiply 22 by 484 which will give a cube of 22 as 10648. It’s beneficial to remember the square of the first 50 numbers.

First 50 Number Cubes

13 = 1	113 = 1331	213 = 9261	313 = 29791	413 = 68921
23 = 8	123 = 1728	223 = 10648	323 = 32768	423 = 74088
33 = 27	133 = 2197	233 = 12167	333 = 35937	433 = 79507
43 = 64	143 = 2744	243 = 13824	343 = 39304	443 = 85184
53 = 125	153 = 3375	253 = 15625	353 = 42875	453 = 91125
63 = 216	163 = 4096	263 = 17576	363 = 46656	463 = 97336
73 = 343	173 = 4913	273 = 19683	373 = 50653	473 = 103823
83 = 512	183 = 5832	283 = 21952	383 = 54872	483 = 110592
93 = 729	193 = 6859	293 = 24389	393 = 59319	493 = 117649
103 = 1000	203 = 8000	303 = 27000	403 = 64000	503 = 125000

Cube Root

Cube Root symbol is ‘∛‘ and it’s called a radical symbol. If we’re trying to find the cube root of a number x then this number is called the radicand. It’s expressed as ∛x.

How to find Cube Roots?

To find the Cubes and Cube Roots of any given number we must check first if the given number is a perfect cube or not. If the given number is a perfect cube number then we can easily find that number with the help of the Prime Factorisation Method. If the number is not a perfect cube number then we can use the Long Division Method to find out the cube root of that number.

Cube root of a number by Long Division Method

Let’s understand the Long Division Method for finding out the cube root of 15,625. Below are the steps to find the cube root of 15625 by the long division method:

Step 1: Separate the digits by placing commas from right to left once in three digits pairs. for example 15, 625.
Step 2: Now, find a number cube that is less than or equal to the first two digits of the number (i.e. 15). For number 15 we find the perfect cube of 2. The cube of 2 is 8 which is less than 15. So in this division: Remainder is 7, Divisor and Quotient are 2.
Step 3: Now we’ll bring down 625, and the new number will be 7625 which new Dividend.
Step 4: Now we’ll find Divisor for 7625, we’ll get the first digit from the last quotient multiplied by 6 (i.e. 6).
Step 5: Now, we’ll find a unit place of the quotient in such a way that (60 x a² x a) + a³. For example, if a is 5 then (60 x 5² x 5) + 5³ = 7625
Step 6: Upon Dividing 7625 by 65 we get the remainder as 0.

Hence, the cube root that we found by long division method is 25.

First 50 Numbers Cube Root

∛1 = 1	∛11 = 2.224	∛21 = 2.759	∛31 = 3.141	∛41 = 3.448
∛2 = 1.259	∛12 = 2.289	∛22 = 2.802	∛32 = 3.175	∛42 = 3.476
∛3 = 1.442	∛13 = 2.351	∛23 = 2.844	∛33 = 3.208	∛43 = 3.503
∛4 = 1.587	∛14 = 2.41	∛24 = 2.884	∛34 = 3.240	∛44 = 3.530
∛5 = 1.710	∛15 = 2.466	∛25 = 2.924	∛35 = 3.271	∛45 = 3.557
∛6 = 1.817	∛16 = 2.52	∛26 = 2.962	∛36 = 3.302	∛46 = 3.583
∛7 = 1.913	∛17 = 2.571	∛27 = 3	∛37 = 3.332	∛47 = 3.609
∛8 = 2	∛18 = 2.621	∛28 = 3.037	∛38 = 3.362	∛48 = 3.634
∛9 = 2.08	∛19 = 2.668	∛29 = 3.072	∛39 = 3.391	∛49 = 3.659
∛10 = 2.154	∛20 = 2.714	∛30 = 3.107	∛40 = 3.420	∛50 = 3.684

Cubes and Cube Roots Use

The application of Cubes and Cube Roots is in solving many Quadratic equations, Trigonometric, and mensuration equations. However, finding Cubes and Cube Roots are the basic building blocks of mathematics, and having expertise in number calculations will always be helpful in solving complex questions.

Memorizing Cubes and Cube Roots to 50 numbers are very essential so that we don’t waste time-solving Cubes and Cube Roots.

Learn Surds and Indices which have the direct application of Cubes and Cube Roots.

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