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.
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 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.
How to find the Cube of a number?
Multiply the given number thrice to get a cube of it.
How to find the Cube Root of a number?
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.
What is a perfect cube?
A perfect cube is a cube of any number which is an integer.
Which is the easiest way to find a cube root?
The easiest way to find out the cube root of a perfect cube number is by the prime factorization method.
Explain the Prime factorization method for cube root?
Prime Factorize the number first and if the factors of the number can equally be grouped in triples, then that number is a perfect cube.