Cube identities: Class 9 practice
(a + b)³ = a³ + b³ + 3ab(a + b) and (a − b)³ = a³ − b³ − 3ab(a − b). Read the method, follow the three worked examples, then try the six questions.
What you should know first
- Squares and cubes
- Expanding brackets
The idea behind cube identities
(a + b)³ = a³ + b³ + 3ab(a + b) and (a − b)³ = a³ − b³ − 3ab(a − b). When a + b + c = 0, a³ + b³ + c³ = 3abc.
Step-by-step method
- Spot which identity fits.
- Identify a, b (and c).
- Substitute and evaluate.
Worked example 1 (Basic)
Use an identity to find 103³.
- (100 + 3)³ = 100³ + 27 + 3 × 100 × 3 × 103
- = 1092727
- Answer: 1092727
Worked example 2 (Standard)
Use an identity to find 99³.
- (100 − 1)³ = 100³ − 1 + 3 × 100 × -1 × 99
- = 970299
- Answer: 970299
Worked example 3 (Challenge)
Without cubing each number, find (-4)³ + (7)³ + (-3)³.
- -4 + 7 + -3 = 0
- So the sum = 3 × -4 × 7 × -3 = 252
- Answer: 252
Common mistakes and misconceptions
- Writing (a + b)³ = a³ + b³
- Missing the 3abc shortcut
Practice questions
- (Basic) Use an identity to find 105³.
- (Basic) Use an identity to find 104³.
- (Standard) Use an identity to find 98³.
- (Standard) Use an identity to find 95³.
- (Challenge) Without cubing each number, find (6)³ + (7)³ + (-13)³.
- (Challenge) Without cubing each number, find (-1)³ + (6)³ + (-5)³.
Answer key with full working
- Q1: (100 + 5)³ = 100³ + 125 + 3 × 100 × 5 × 105 → = 1157625 → Answer: 1157625
- Q2: (100 + 4)³ = 100³ + 64 + 3 × 100 × 4 × 104 → = 1124864 → Answer: 1124864
- Q3: (100 − 2)³ = 100³ − 8 + 3 × 100 × -2 × 98 → = 941192 → Answer: 941192
- Q4: (100 − 5)³ = 100³ − 125 + 3 × 100 × -5 × 95 → = 857375 → Answer: 857375
- Q5: 6 + 7 + -13 = 0 → So the sum = 3 × 6 × 7 × -13 = -1638 → Answer: -1638
- Q6: -1 + 6 + -5 = 0 → So the sum = 3 × -1 × 6 × -5 = 90 → Answer: 90
Syllabus note
This skill is listed under "Polynomials" for Class 9 in the NCERT/CBSE books. ICSE, IGCSE, IB and state boards may teach it in a different chapter or year.
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.
Related practice and notes
Frequently asked questions
Which chapter is cube identities in?
"Polynomials", Class 9 Maths (NCERT/CBSE). Other boards may place it differently.
How were the answers checked?
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.