Cubes and Cube Roots - REET Maths Notes
Cubes and Cube Roots in REET Maths covers interesting facts like the Hardy-Ramanujan number alongside straightforward calculations like the cube root of a fraction. This page clarifies both sides with fully worked examples.
- 2 questions in REET 2024
- Class 8 level
- 6 min read
How REET asks this topic
REET 2024 had one question on simplifying a fraction (7/875) and finding its cube root, and another factual question on the famous Hardy-Ramanujan number 1729's special property. So both direct calculation and a few interesting number facts need to be remembered here.
Cubes from 1 to 15
| Number | Cube | Number | Cube |
|---|---|---|---|
| 1 | 1 | 9 | 729 |
| 2 | 8 | 10 | 1000 |
| 3 | 27 | 11 | 1331 |
| 4 | 64 | 12 | 1728 |
| 5 | 125 | 13 | 2197 |
| 6 | 216 | 14 | 2744 |
| 7 | 343 | 15 | 3375 |
| 8 | 512 |
For calculation speed in REET, it's useful to memorise the cubes from 1-12.
The pattern in the units digit of cubes
The units digit of each digit's (0-9) cube is fixed - 1^3=1, 2^3=8, 3^3=27 (->7), 4^3=64 (->4), 5^3=125 (->5), 6^3=216 (->6), 7^3=343 (->3), 8^3=512 (->2), 9^3=729 (->9), 0^3=0. Notice that 2 and 8, and 3 and 7, swap their units digits when cubed - this pattern helps you find cube roots quickly.
To find a cube root, split the number into triples of prime factors - for example, 1728 = 2^6 x 3^3 = (2^2x3)^3 = 12^3, so cube root of 1728 = 12.
Another use of this pattern: if a large perfect cube number's units digit is 7, its cube root will end in the units digit 3 - for example, 2197 ends in 7, so its cube root's units digit is 3, and indeed cube root of 2197 = 13, since 13^3=2197.
Worked example: cube root of a fraction
REET 2024's question asked for the cube root of 7/875. First factorise 875: 875 = 125x7 = 5^3x7, so 7/875 = 7/(5^3x7) = 1/5^3 = 1/125. The cube root of 1/125 is then 1/5, since 5^3=125.
The same approach applies to decimals - for example, to find the cube root of 0.008, write 0.008 = 8/1000 = 2^3/10^3, so the cube root is 2/10 = 0.2. Writing a fraction or decimal in its simplest form first is the most reliable way to find a cube root.
Negative numbers also have real cube roots. Example: find the cube root of -216. Since 216=6^3, and the cube root of a negative number is negative, the cube root of -216 is -6, because (-6)^3=-216.
The Hardy-Ramanujan number and sums of cubes
- Hardy-Ramanujan number
- 1729 is the smallest number that can be written as the sum of two cubes in two different ways: 1729 = 1^3+12^3 = 9^3+10^3.
- Check
- 1^3+12^3 = 1+1728 = 1729; 9^3+10^3 = 729+1000 = 1729 - both equal.
- Sum of the cubes of the first n natural numbers
- Formula: [n(n+1)/2]^2. Example: for n=5, sum = (5x6/2)^2 = 15^2 = 225.
- Another example
- 4104 can also be written as the sum of two cubes in two ways: 4104 = 2^3+16^3 = 9^3+15^3.
Cubes and volume - a practical connection
A cube number is directly connected to the volume of a cube-shaped object - the volume of a cubical box or block = side^3. This is the most natural link between geometry and cube numbers.
For example, if a cubical box has a side of 5 centimetres, its volume = 5^3 = 125 cubic centimetres. Similarly, if a cubical tank has a volume of 216 cubic metres, its side = cube root of 216 = 6 metres. REET can combine geometry and cube roots in this kind of practical question.
A useful identity involving cubes
- Identity
- a^3+b^3+c^3-3abc = (a+b+c)(a^2+b^2+c^2-ab-bc-ca)
- Check (a=1,b=2,c=3)
- Left side: 1+8+27-3x1x2x3=36-18=18. Right side: (1+2+3)(1+4+9-2-6-3)=6x3=18 - both equal.
- Special case
- If a+b+c=0, then a^3+b^3+c^3=3abc - this result is useful in algebra questions in higher classes.
How to teach this to children
Classroom activity
In Class 8, use unit-cube blocks to physically build models of size 1^3, 2^3, 3^3 - have children count and see why 2x2x2=8 and 3x3x3=27 blocks are needed, which makes the real meaning of the word 'cube' clear.
Teaching aids
Put up a chart of cubes from 1-12 and a units-digit pattern table in class, so children learn to quickly estimate cube roots.
Common misconception
Children often mistake 'cube' (n^3) for 'multiply by 3' (3n). This misconception is most effectively cleared up using physical block models.
Assessment idea
Have students practise picking out perfect cube numbers from a given list, and ask how each one's cube root's units digit can be found instantly from the pattern.
Cross-curricular link
In science class, connect cube numbers to calculating the volume of a cubical container or box - this reinforces learning across both maths and science, and children come to see the concept as connected rather than isolated.
Trick to remember / exam tip
To find the cube root of a large perfect cube quickly, split the number into groups of 3 digits from the right. The units digit of the right-hand group gives the cube root's units digit (from the pattern table), and the left-hand group gives an estimate of the cube root's tens digit - for example, in 1728, the right group '728' (units digit 8, so cube root units digit 2) and the left part '1' (the next cube above 1^3=1 is 2^3=8, so tens digit is 1), giving cube root = 12.
Questions asked in REET 2024 (with solutions)
Question 120
The value of ∛(7/875) is :
- A1/15
- B1/4
- C1/7
- D1/5
Explanation
7/875 को सरल करने पर 875 = 7 × 125, अतः 7/875 = 1/125। 1/125 का घनमूल = 1/5, क्योंकि 5³ = 125। इसलिए (D) सही है।
Question 96
The smallest number that can be expressed as a sum of two cubes in two different ways in the followings
- A1729
- B1728
- C1331
- D1332
Explanation
1729 को 'हार्डी–रामानुजन संख्या' कहते हैं: 1729 = 1³ + 12³ = 9³ + 10³। यह सबसे छोटी संख्या है जिसे दो घनों के योग के रूप में दो भिन्न प्रकार से लिखा जा सकता है। 1728 = 12³ और 1331 = 11³ स्वयं पूर्ण घन हैं।
Practice Questions
Question 1
What is the value of 6³?
- A196
- B206
- C216
- D226
Explanation
6x6x6=216.
Question 2
What is the value of ∛343?
- A6
- B7
- C8
- D9
Explanation
7x7x7=343, so cube root of 343 = 7.
Question 3
Which of the following is not a perfect cube?
- A125
- B216
- C300
- D512
Explanation
125=5^3, 216=6^3, 512=8^3 are perfect cubes, but 300 is not the cube of any integer (6^3=216, 7^3=343, and 300 falls in between).
Question 4
If the unit digit of a perfect cube is 2, what is the unit digit of the original number whose cube it is?
- A2
- B4
- C8
- D9
Explanation
By the cube-units-digit pattern, 8^3=512 (gives units digit 2), so the original number's units digit is 8.
Question 5
What is the value of 2³ + 3³ + 4³?
- A89
- B99
- C109
- D72
Explanation
8+27+64=99.
Question 6
What is the sum of the cubes of the first 4 natural numbers?
- A90
- B100
- C110
- D64
Explanation
By the formula [n(n+1)/2]^2, for n=4: (4x5/2)^2=10^2=100. Check: 1+8+27+64=100.
Question 7
What is the value of ∛(0.027)?
- A0.3
- B0.03
- C3
- D0.9
Explanation
0.3x0.3x0.3=0.027, so cube root of 0.027 = 0.3.
Question 8
What is the value of ∛(64/729)?
- A8/27
- B4/9
- C2/3
- D16/81
Explanation
64=4^3 and 729=9^3, so cube root of 64/729 = 4/9.
Question 9
What is the smallest number by which 72 must be multiplied to make the product a perfect cube?
- A2
- B3
- C6
- D9
Explanation
72=2^3x3^2. To make it a cube, 3^2 needs to become 3^3, meaning one more factor of 3 is needed. 72x3=216=6^3.
Question 10
Which is the most suitable teaching aid to concretely explain cube numbers to Class 8 students?
- Aयूनिट क्यूब (इकाई घनों) से n×n×n आकार के घन मॉडल बनवाना
- Bकेवल सूत्र बोर्ड पर लिखना
- Cघर पर स्वयं पढ़ने के लिए छोड़ देना
- Dकेवल मौखिक व्याख्या देना
Explanation
Building models with physical unit-cube blocks is the most effective way for children to directly grasp the three-dimensional concept of a 'cube'.
Frequently Asked Questions
How many questions come from Cubes & Cube Roots in REET?+
REET 2024 had 1-2 questions from this topic in the Maths section, based on cube roots of fractions and the Hardy-Ramanujan number.
What is the Hardy-Ramanujan number?+
1729 is the smallest number that can be written as the sum of two cubes in two different ways: 1^3+12^3 and 9^3+10^3.
What is the main method for finding a cube root?+
The prime factorisation method, in which the number is split into triples (groups of three) of prime factors.
What is the formula for the sum of the cubes of the first n natural numbers?+
[n(n+1)/2]^2 - the square of the sum of the first n numbers.
What's the relationship between a cube and a cube root?+
A cube root is the inverse operation of cubing - if a^3=b, then the cube root of b is a. The two operations cancel each other out.
Does every number have a whole-number cube root?+
No, only perfect cube numbers (like 8, 27, 64) have whole-number cube roots; other numbers can have irrational cube roots.
Is the cube root of a negative number possible?+
Yes, unlike square roots, the cube root of a negative number is also real - for example, the cube root of -8 is -2, since (-2)^3 = -8.
What's a quick way to check if a number is a perfect cube?+
Break the number into its prime factors - if the power of every prime factor is a multiple of 3 (3, 6, 9...), the number is a perfect cube.
Are 1 and 0 perfect cube numbers?+
Yes, since 1^3=1 and 0^3=0, both are considered perfect cube numbers - they're the two smallest perfect cubes.
What's the basic difference between a square and a cube?+
A square (n^2) is obtained by multiplying a number by itself once, while a cube (n^3) is obtained by multiplying it by itself twice - a cube has one extra multiplication.
Are there other numbers besides 1729 that can be written as the sum of two cubes in two ways?+
Yes, such numbers are called taxicab numbers. 1729 is the smallest such number, and 4104 (=2^3+16^3=9^3+15^3) is the next one.
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- 17 September 2026 — English page published — mirrors the fact-checked Hindi page, independently bilingual-accuracy checked