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.
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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|
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.
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.
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