Powers and Exponents is an integral part of Mathematical Arithmetic. It is extensively taught to students in middle school, giving them insight into concepts like squares, cubes, and their roots. These concepts are of high utility not just in understanding maths for competitive exams, but also in day to day life, forming a crucial component of mental mathematics. Not to forget, ability tests like JAT, CAT, and GMAT include Perfect Squares and Perfect Cubes in their Quantitative Aptitude section. This article focuses specifically on Perfect Cubes and its list, discussing the common types of questions associated with it and how once can solve them without any hassle.
This Blog Includes:
- What are Perfect Cubes?
- List of Perfect Cubes
- How to Find the Perfect Cube?
- Application of Perfect Cube
- Useful Tricks and Methods
- Sample Questions and Solutions
- Perfect Cubes Questions for Practice
- [BONUS] How to Work Around Perfect Cubes in GMAT?
What are Perfect Cubes?
Perfect Cubes can be understood as whole numbers which arise by multiplying a number with itself thrice. It is important to focus on the word ‘whole numbers’ as this is the key distinction between Cubes and Perfect Cubes.
- These numbers are a key part of mathematical studies, used not just in Algebra, but mensuration as well to find volumes and other 3-dimensional values.
- All numbers have a Cubic value associated with themselves, but only integers have Perfect Cubic value, meaning that they have no decimal values.
- Questions relating to Perfect Cubes generally involve basic arithmetic operations, which require the students to multiply or divide some value in the given number to make it a Perfect Cube.
- Occasionally, questions which require students to identify the Perfect Cube from a host of given numbers also appear.
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List of Perfect Cubes
These numbers have a distinction over squares, as they do not change the negative value of the number. Students can try to memorise the cubes of numbers up to 10, as it would greatly help increase their speed in solving maths quiz. Here is a list of Perfect cubes, along with their roots:
|Number||Perfect Cubes||Number||Perfect Cubes|
Short Maths Tricks
How to Find the Perfect Cube?
You can find the perfect cube through this simple method:
- Prime factorize the number starting from the smallest prime number (2)
- Once you are done with the prime factorization, club all the same factors together in a group of 3
- Repeat the step for all sets of group of three factors
- If there is any number left behind after setting the groups, the number is not a perfect cube
- If there is no other number left, the number is a perfect cube
Application of Perfect Cube
A cube is a three-dimensional figure that has all equal sides. The volume of a cube is defined by the product of its dimensions. Since the cube’s dimensions are the same, the volume of the cube will be a*a*a i.e. a^3 cubic unit.
For example, if the dimension of a cube is 5, then according to that, 5*5*5 is 125. Hence, 125 is the volume of the cube.
Useful Tricks and Methods
Generally, Perfect Cubes questions are direct in nature, requiring simple Arithmetic operations to arrive at the correct solution. However, there are some useful tricks and directions which you can apply while solving such questions. Here are a few of them:
- You should start off with factoring the number in question. These factors must be prime numbers, having no further factors within themselves. Put the same prime numbers together, making groups of 3 with them.
- Always remember to cross-check the question by verifying your factorisation process.
- In case of multiple-choice questions, evaluate the options before jumping straight into mathematical operations. This can help you save time.
Sample Questions and Solutions
To help you understand the methodology of such questions in a better way, here are some solved questions on Perfect Cubes:
Q1: What is the least number which must be multiplied with 38250 to make it a perfect cube?
Ans: To solve this question, follow the given steps:
- Divide the number into its prime factors. For 38250, prime factors will be: 2x5x5x5x3x3x17
- Put these numbers into groups of 3, as (2), (5,5,5), (3,3), (17).
- Note that only 5 has a complete set of numbers. 2, 3, and 17 have some missing number of factors with them.
- Identify the missing factors. For this question, it would be (2,2), (3), (17,17). Multiply these missing factors together.
- The resultant number (3468) is your answer. The number 38250 must be multiplied with 3468 to make it a perfect cube.
Q2: Which of these numbers is a perfect cube?
Ans: To solve multiple-choice, perfect cubes problem of this type, it is always advisable to evaluate the choices using mental maths before performing mathematical calculations. Here is the procedure to solve this question:
- Determine the approximate cube root range of the greatest and smallest number in the options. Two steps of classic cube root operation will give us the range of numbers, which in this case lies between 30 and 40.
- Note digits of all options at unit value. Since only even numbers can have even multiples, the cube root must have an even number at its end.
OPTION A: (39316) ends with 6. As we know, only the numbers ending with 6 have their cube ending with a 6. Hence, we can arrive at the conclusion that only 36 fits in our range of options. When the resultant of the hit and trial method is in close proximity to or greater than 39316, we would discard this option.
OPTION B: (27006) is very close to the cube of 30 (27000) to be a cube of any other whole number. Hence, it can be discarded.
OPTION C: (46658) ends with 8, implying a 2 at the end of the cube root value. This, combined with the first step suggests that the probable number is 32, which is not the case. Hence, option c is discarded.
OPTION D ends with 8, which also points towards 32 as the probable root. Upon solving, we can arrive at the conclusion that option d is correct.
GMAT Sample Paper
Series and Sequence Questions for GMAT
Perfect Cubes Questions for Practice
Now, here are some practice questions for you to attempt and sharpen your skills at understanding Perfect Cubes. Although GMAT questions are generally multiple-choice, questions without choice would help you increase your confidence as well as make you thorough with all the steps involved.
Q1: In order to make 4638 a Perfect Cube, what is the least number which must be multiplied by it?Q2: Which of these numbers is a perfect cube?
Q3: What is the perfect cube value that lies between 5000 and 6000?
Q4: What is the least number that must be multiplied by 3102 to make it a Perfect Cube?
Q5: What is the only Perfect Cube lying between 8001 and 10000?
Q6: What is the cube root of 1?
Q7: Find the cube root of 54/250.
Q8: Evaluate the value of (3.5)³
Q9: Is 1331 a perfect cube?
Q10: Find out the square root of 5324.
Inequality Questions for GMAT
Mean Median Mode Questions
[BONUS] How to Work Around Perfect Cubes in GMAT?
Hope this article was useful to you in understanding the concept of Perfect Cubes and how to solve questions relating to it. If you are confused about any more such topics that appear in competitive exams like GMAT, we are here to help. At Leverage Edu, we offer standardised test prep tools to make you proficient in all aspects of these tests. What’s more, we also offer a host of career guidance and foreign admission facilities to give your career the wings it deserves. Book a free e-meeting to avail this opportunity today!
- Prime factorize the number starting from the smallest prime number (2)
- Once you are done with the prime factorization, club all the same factors together in a group of 3.
- Repeat the step for all sets of group of three factors.
The values of cubes from 1 to 20 are 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728, 2197, 2744, 3375, 4096, 4913, 5832, 6859 and 8000.How to learn cubes from 1 to 50? ›
To calculate the value of cubes from 1 to 50 we can use the following method: Multiplication by itself: In this method, the number is multiplied thrice and the resultant product gives us the cube of that number. For example, the cube of 8 = 8 × 8 × 8 = 512.What is the perfect cube lesson? ›
Activity 2: Perfect Cubes
A perfect cube is the cube of a whole number, found by multiplying a whole number by itself three times. For example, the number 8 is a perfect cube because 2 × 2 × 2 = 8 .
In order to check whether a number is a perfect cube or not, we find its prime factors and group together triplets of the prime factors. If no factor is left out then the number is a perfect cube. However if one of the prime factors is a single factor or a double factor then the number is not a perfect cube.What is a perfect cube example? ›
What Are Perfect Cubes. When we multiply the same integer thrice, the number we get is called a perfect cube number. For example, if we multiply the number 2 thrice, the product is 8. Therefore, 8 is a perfect cube.How many 100 perfect cubes are there? ›
So we can say that the positive integers whose cubes are 100 or below are 1, 2, 3 and 4 only. Thus there are 4 perfect cubes from 1 to 100 and they are 1, 8, 27 and 64. Thus there are 4 perfect cubes from -100 to -1.Why is 29 not a cube number? ›
The number 29 is prime. Here, the prime factor 29 is not in the power of 3 and this implies that the cube root of 29 is irrational, hence 29 is not a perfect cube.Is cubes a good math strategy? ›
The CUBES math strategy is a great tool for students to have to help successfully solve story problems.What is the shortcut method of cubes? ›
The easy way to find the Cube of a Two-Digit Number is by using (a + b)³ = a³ + 3a²b + 3ab² + b³.
Flashcards are a great no-fuss way to learn math facts. Shuffling a deck makes it easy to mix up the order in which students practice, a learner can study them together with a tutor, a peer, or on their own, and flashcards provide both visual stimuli and a chance for kinesthetic learning.What is the math trick for square? ›
- 2 x 3 = 6. After the 6 we add the number 25:
- 40 x 41 = 1,640. We add 25 on to the end of the number:
- 47 – 25 = 22. The difference between 50 and 47 is:
- 50 – 47 = 3. We square the number 3:
- 56 – 25 = 31. ...
- 56 – 50 = 6.
Finding the Square of a Number is a simple method. We need to multiply the given number by itself to find its square number. The square term is always represented by a number raised to the power of 2. For example, the square of 6 is 6 multiplied by 6, i.e., 6×6 = 62 = 36.Which cube should I learn first? ›
It is better to start other events like 2x2 and Big cubes (4x4-7x7). Or if you want to start off with smaller cubes then better go with 2x2 and Pyraminx or Skewb. The best-recommended order is to go with 2x2 and 4x4, and later on with pyraminx.Why is 27 a perfect cube? ›
So, the cube root is cancelled by the cube of 3. Hence, the value of the cube root of 27 is equal to 3. Also, we can say that 27 is a perfect cube number since its cube root value is a whole number.What is perfect cube vocabulary? ›
A perfect cube is any number that can be expressed as the product of three equal integers. In other words, a perfect cube is a number that can be made by cubing an integer. Every perfect cube is itself an integer. Also, the perfect cube will have the same sign as the number you started with.What is the formula for finding the cube? ›
The volume of a cube is obtained by multiplying its side three times. The formula of volume of the cube can thus be given as, Volume of cube = s3, where s is the side length of the cube.What does 7 cubed mean? ›
7 Cubed. = 73. = 7 × 7 x 7.How many perfect cubes are there between 1 and 1000 how to find? ›
There are 8 perfect cubes between 1 and 1000.What is perfect cube example problems? ›
Perfect cube numbers can be obtained by multiplying every number thrice by itself. For example, 1 × 1 × 1 = 1 and 2 × 2 × 2 = 8 and so on. The list of perfect cubes from 1 to 10 is as follows: 1, 8, 27, 64, 125, 216, 343, 512, 729, and 1000.
Answer: 3757 is not a perfect square number because it has the unit digit 7.What are the properties of a perfect cube? ›
- If a number ends in 0, its cube ends in 0.
- If a number ends in 2, its cube ends in 8.
- If a number ends in 3, its cube ends in 7.
- If a number ends in 4, its cube ends in 4.
- If a number ends in 5, its cube ends in 5.
- If a number ends in 6, its cube ends in 6.
Cube shaped objects include sugar cubes, rolling dice, ice cubes, and the world-famous Rubik's Cube!What are the first 5 perfect cube numbers? ›
The first few are 1, 8, 27, 64, 125, 216, 343, ... (OEIS A000578).Is 512 a perfect cube? ›
Since 512 is a perfect cube, we will use here the prime factorisation method, to get the cube root easily. Learn more here, to calculate the value of 3√512.Which all numbers are perfect cubes? ›
|Number (x)||Multiplied Three times by itself||Cubes (x3)|
|2||2× 2× 2||8|
|3||3× 3× 3||27|
|4||4× 4× 4||64|
|5||5× 5× 5||125|
In order for a number to be a perfect square and a perfect cube, the sixth power of the number needs to be less than 1,000,000. There are ten such numbers 1, 64, 729, 4,096, 15,625, 46,656, 117,649, 262,144, 531,441, and 1,000,000.What is the ZZ method? ›
The ZZ method (short for "Zbigniew Zborowski") is a modern speedcubing method originally proposed by Zbigniew Zborowski in 2006. The method was designed specifically to achieve high turning speed by focusing on move ergonomics, and is the combination of a block-building method and a layer-by-layer method.What is the hardest cube algorithm? ›
The superflip is often dubbed the hardest scramble. It is basically the solved cube but with all edges flipped.What is the hardest cube solver? ›
Mastermorphix Rubik's Cube
What is this? The standard Mastermorphix puzzle is very hard, which is why it is on our list of the hardest twisty puzzles. It is a 33 Rubik's Cube with a different shape that is much harder to solve.
However, the world record average Sqaure-1 time is 5.02 seconds, claimed by Max Siauw (USA) on 29 May 2022 in Boston, Massachusetts, USA.Is a 2x2 cube easier than a 3x3? ›
Introduction. The 2x2x2 Rubik's cube, or in its official name- the Pocket Cube, is another puzzle in the Rubik's cube series, invented by Erno Rubik. It is considered the "easy" version of the Rubik's cube. You will find out that solving the 2x2 cube is much easier than solving the classic 3x3x3 cube.Is 4x4 harder than 5x5? ›
A big misconception regarding these puzzles is that the higher order you go, the more difficult it becomes. After learning the 4x4 and 5x5 cubes; this is effectively not true because although more pieces must be solved, the process in which you solve them is the exact same as a 4x4 or 5x5.Why is 200 not a perfect cube? ›
Is 200 a Perfect Cube? The number 200 on prime factorization gives 2 × 2 × 2 × 5 × 5. Here, the prime factor 5 is not in the power of 3. Therefore the cube root of 200 is irrational, hence 200 is not a perfect cube.Why 648 is not a perfect cube? ›
After grouping the prime factors in triplets, one factor 3 is left without grouping. Thus, 648 is not a perfect cube.Why is 56 not a cube number? ›
Is 56 a Perfect Cube? The number 56 on prime factorization gives 2 × 2 × 2 × 7. Here, the prime factor 7 is not in the power of 3. Therefore the cube root of 56 is irrational, hence 56 is not a perfect cube.What is the rule of perfect cubes? ›
A perfect cube of a number is a number that is equal to the number, multiplied by itself, three times. If x is a perfect cube of y, then x = y3. Therefore, if we take the cube root of a perfect cube, we get a natural number and not a fraction. Hence, 3√x = y.Are there only 20 perfect cubes from 1 to 1000? ›
Answer: There are 8 perfect cubes between 1 and 1000.What are the first 100 perfect cubes? ›
So we can say that the positive integers whose cubes are 100 or below are 1, 2, 3 and 4 only. Thus there are 4 perfect cubes from 1 to 100 and they are 1, 8, 27 and 64. Thus there are 4 perfect cubes from -100 to -1.What is perfect cube equation examples? ›
A number is a perfect cube when it can be factored into an integer with an exponent with the value of 3. For example, 8 is a perfect cube because 2^3 = 8 , or 2 * 2 * 2 = 8.
Is 729 a perfect cube? 3√729 = 9 and 9 is an integer. So, 729 is a perfect cube.Is the square root of 8 a perfect square? ›
The square root of 8 in radical form is represented as √8 which is also equal to 2√2 and as a fraction, it is equal to 2.828 approximately. Squares root of a number is the number, which, on multiplying by itself gives the original number. Since 8 is not a perfect square, hence the value is represented in root form.Is 250 a perfect cube if not find the smallest? ›
∴ The smallest number by which 250 should be divided to make it a perfect cube is 2.Is 49 a perfect square? ›
Informally: When you multiply an integer (a “whole” number, positive, negative or zero) times itself, the resulting product is called a square number, or a perfect square or simply “a square.” So, 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, and so on, are all square numbers.Is 27 a perfect square? ›
The number 27 is lies between 25 and 36, therefore 27 is not a perfect square of an integer.What is the greatest 1 digit perfect cube? ›
In the five digit number 1b6a3, a is the greatest single digit perfect cube and twice of it exceeds by 7.