Exercise 11.1 — Monomial x Monomial
Problems based on multiplying a monomial by another monomial.
What is Exercise 11.1 About?
Exercise 11.1 of Class 8 Mathematics (CBSE, Telangana & Andhra Pradesh syllabus) focuses on the multiplication of algebraic monomials. A monomial is a single-term algebraic expression such as 6, 7k, -3l, or 5x²y. When you multiply two or more monomials, you multiply their coefficients (numbers) separately and their variable parts separately using the laws of exponents.
This exercise contains seven types of problems: multiplying pairs of monomials, completing a multiplication table, finding volumes of rectangular boxes, multiplying multiple monomials, evaluating expressions with substitution, and creating your own monomials. All of these build the foundation needed for polynomial multiplication in Class 9 and 10.
Question 1 — Multiplying Pairs of Monomials
Each sub-question asks you to find the product of two monomials. The method is always the same: group the numbers and group the variables, then simplify each group.
Since 6 is a plain number (no variable), we simply multiply 6 × 7 = 42 and attach the variable k.
Sign rule reminder: Negative × Negative = Positive. So (−3) × (−2) = +6.
t² × t² = t²⁺² = t⁴Question 2 — Complete the Multiplication Table
This problem gives a large multiplication table with row and column headers as monomials. Each cell contains the product of the row-header monomial and the column-header monomial. Two sample cells are pre-filled to guide you (e.g., 3x × 5x = 15x²; −2x² × 5x = −10x³). The table below shows the complete, filled version.
| × | 5x | −2y² | 3x² | 6xy | 3y² | −3xy² | 4xy² | x²y² |
|---|---|---|---|---|---|---|---|---|
| 3x | 15x² | −6xy² | 9x³ | 18x²y | 9xy² | −9x²y² | 12x²y² | 3x³y² |
| 4y | 20xy | −8y³ | 12x²y | 24xy² | 12y³ | −12xy³ | 16xy³ | 4x²y³ |
| −2x² | −10x³ | 4x²y² | −6x⁴ | −12x³y | −6x²y² | 6x³y² | −8x³y² | −2x⁴y² |
| 6xy | 30x²y | −12xy³ | 18x³y | 36x²y² | 18xy³ | −18x²y³ | 24x²y³ | 6x³y³ |
| 2y² | 10xy² | −4y⁴ | 6x²y² | 12xy³ | 6y⁴ | −6xy⁴ | 8xy⁴ | 2x²y⁴ |
| 3x²y | 15x³y | −6x²y³ | 9x⁴y | 18x³y² | 9x²y³ | −9x³y³ | 12x³y³ | 3x⁴y³ |
| 2xy² | 10x²y² | −4xy⁴ | 6x³y² | 12x²y³ | 6xy⁴ | −6x²y⁴ | 8x²y⁴ | 2x³y⁴ |
| 5x²y² | 25x³y² | −10x²y⁴ | 15x⁴y² | 30x³y³ | 15x²y⁴ | −15x³y⁴ | 20x³y⁴ | 5x⁴y⁴ |
Question 3 — Volume of Rectangular Boxes
The volume of a rectangular box (cuboid) is calculated using the formula V = l × b × h. When the dimensions are given as algebraic monomials, you simply multiply all three monomials together. This question tests your ability to apply monomial multiplication in a real-world geometric context.
Volume (V) = Length (l) × Breadth (b) × Height (h)| S.No. | Length (l) | Breadth (b) | Height (h) | Volume = l × b × h |
|---|---|---|---|---|
| (i) | 3x | 4x² | 5 | 3x × 4x² × 5 = 60x³ |
| (ii) | 3a² | 4 | 5c | 3a² × 4 × 5c = 60a²c |
| (iii) | 3m | 4n | 2m² | 3m × 4n × 2m² = 24m³n |
| (iv) | 6kl | 3l² | 2k² | 6kl × 3l² × 2k² = 36k³l³ |
| (v) | 3pr | 2qr | 4pq | 3pr × 2qr × 4pq = 24p²q²r² |
Let's trace through problem (iv) in detail so the method is clear:
Question 4 — Products of Multiple Monomials
Here, you multiply three or more monomials at once. The principle is the same — but you need to be careful to collect all coefficients and all powers of each variable correctly.
Questions 5 & 6 — Evaluate Expressions by Substitution
These questions assign algebraic expressions to variables and ask you to calculate the value of a combined expression. You first find the product of the assigned expressions, then simplify.
Question 7 — Create Your Own Monomials and Multiply
This open-ended question encourages you to explore monomial multiplication on your own. You pick any monomials you like, then find their product. The textbook uses 2a, 3a²b, −4b³ as an example:
You can try monomials involving three or more variables, or choose all-positive terms for practice. The key is to demonstrate that you can correctly group and add exponents for each variable.
- 5x, 2x²y, 3y² → product = 30x³y³
- −2p², 3pq, −q² → product = 6p³q³
- 4ab, 2bc, 3ca → product = 24a²b²c²
Common Mistakes to Avoid
- Forgetting to add exponents: When multiplying
x² × x³, students sometimes writex⁶(multiplied) instead of the correctx⁵(added). Always add exponents for the same base. - Sign errors with negatives: Two negatives multiply to a positive. Three negatives multiply to a negative. Count the number of negative signs before assigning the final sign.
- Mixing up different variables: In a product like
x²y × xy², keep x-powers and y-powers separate. Result: x²⁺¹ × y¹⁺² = x³y³. - Leaving coefficients out: When copying a monomial, students sometimes write only the variable part and drop the number coefficient. Always multiply all the number parts together.
- Confusing addition and multiplication rules: In addition, you can only combine like terms (e.g., 3x + 2x = 5x). In multiplication, unlike terms are fine — 3x × 2y = 6xy.
Quick Reference — All Answers at a Glance
| Question | Expression | Answer |
|---|---|---|
| Q1(i) | 6 × 7k | 42k |
| Q1(ii) | −3l × −2m | 6lm |
| Q1(iii) | −5t² × −3t² | 15t⁴ |
| Q1(iv) | 6n × 3m | 18mn |
| Q1(v) | −5p² × −2p | 10p³ |
| Q3(i) | 3x × 4x² × 5 | 60x³ |
| Q3(ii) | 3a² × 4 × 5c | 60a²c |
| Q3(iii) | 3m × 4n × 2m² | 24m³n |
| Q3(iv) | 6kl × 3l² × 2k² | 36k³l³ |
| Q3(v) | 3pr × 2qr × 4pq | 24p²q²r² |
| Q4(i) | xy · x²y · xy · x | x⁵y³ |
| Q4(ii) | a · b · ab · a³b · ab³ | a⁶b⁶ |
| Q4(iii) | kl · lm · km · klm | k³l³m³ |
| Q4(iv) | pq · pqr · r | p²q²r² |
| Q4(v) | −3a × 4ab × −6c × d | 72a²bcd |
| Q5 | ABC = xy · yz · zx | x²y²z² |
| Q6 | PTR/100 | x³y |
What This Exercise Prepares You For
Fluency in multiplying monomials is the essential building block for the rest of Chapter 11. The next section on multiplying a monomial by a polynomial (Exercise 11.2) extends this exact skill — instead of single terms, you distribute the monomial across each term in a bracket.
These skills also reappear directly in Chapter 12 — Factorisation, where you reverse the process: splitting a product back into its factors. In Class 9 and 10, monomial multiplication is the engine behind expanding brackets in polynomial expressions and verifying identities in algebra.
For Telangana and Andhra Pradesh board exams, Exercise 11.1 type questions regularly appear as 1-mark or 2-mark fill-in-the-table problems. Mastering the product table format (Question 2) gives you a significant advantage.