GSEB Class 8 Maths Solutions Chapter 9 Algebraic Expressions and Identities Exercise 9.2

Question 1. Find the product of the following pairs of monomials?
1. \( 4, 7p \)
2. \( -4p, 7p \)
3. \( -4p, 7pq \)
4. \( 4p^3, -3p \)
5. \( 4p, 0 \)
Answer:
1. To find the product of \( 4 \) and \( 7p \):
\( 4 \times 7p = (4 \times 7) \times p = 28p \)

2. To find the product of \( -4p \) and \( 7p \):
\( -4p \times 7p = (-4 \times 7) \times p \times p = -28p^2 \)

3. To find the product of \( -4p \) and \( 7pq \):
\( -4p \times 7pq = (-4 \times 7) \times p \times pq = -28 \times p^2q = -28p^2q \)

4. To find the product of \( 4p^3 \) and \( -3p \):
\( 4p^3 \times (-3p) = (4 \times (-3))p^3 \times p = -12 \times p^4 = -12p^4 \)

5. To find the product of \( 4p \) and \( 0 \):
\( 4p \times 0 = 0 \)
In simple words: To multiply monomials, first multiply the number parts (coefficients) together. Then, multiply the variable parts (letters) together, remembering to add their exponents if the bases are the same.

Exam Tip: Always remember that when you multiply a term by zero, the result is always zero, regardless of the term's complexity.

Question 2. Find the areas of rectangles with the following pairs of monomials as their lengths and breadths respectively?
(p, q); (10m, 5n); (20x², 5y²); (4x, 3x²); (3mn, 4np)
Answer:
(i) For length \( = p \) and breadth \( = q \):
Area of the rectangle \( = p \times q = pq \)

(ii) For length \( = 10m \) and breadth \( = 5n \):
Area \( = 10m \times 5n = (10 \times 5) \times m \times n = 50mn \)

(iii) For length \( = 20x^2 \) and breadth \( = 5y^2 \):
Area \( = 20x^2 \times 5y^2 = (20 \times 5) \times x^2 \times y^2 = 100x^2y^2 \)

(iv) For length \( = 4x \) and breadth \( = 3x^2 \):
Area \( = 4x \times 3x^2 = (4 \times 3) \times x \times x^2 = 12x^3 \)

(v) For length \( = 3mn \) and breadth \( = 4np \):
Area \( = 3mn \times 4np = (3 \times 4) \times m \times n \times n \times p = 12mn^2p \)
In simple words: To calculate the area of a rectangle, you simply multiply its length by its breadth. When multiplying algebraic terms, multiply the numbers first, then multiply the same variables by adding their powers.

Exam Tip: Remember that the area is always expressed in square units, even when variables are involved. Combine like variables by adding their exponents.

Question 3. Complete the table of products?
Answer:

Exam Tip: Be careful with negative signs when multiplying. An odd number of negative signs in a product results in a negative answer, while an even number results in a positive answer.

Question 4. Obtain the volume of rectangular boxes with the following length, breadth, height respectively?
1. \( 5a, 3a^2, 7a^4 \)
2. \( 2p, 4q, 8r \)
3. \( xy, 2x^2y, 2xy^2 \)
4. \( a, 2b, 3c \)
Answer:
The volume of a rectangular box is found by multiplying its length, breadth, and height.

1. For length \( = 5a \), breadth \( = 3a^2 \), height \( = 7a^4 \):
Volume \( = 5a \times 3a^2 \times 7a^4 = (5 \times 3 \times 7) \times (a \times a^2 \times a^4) = 105a^7 \)

2. For length \( = 2p \), breadth \( = 4q \), height \( = 8r \):
Volume \( = 2p \times 4q \times 8r = (2 \times 4 \times 8) \times p \times q \times r = 64pqr \)

3. For length \( = xy \), breadth \( = 2x^2y \), height \( = 2xy^2 \):
Volume \( = xy \times 2x^2y \times 2xy^2 = (1 \times 2 \times 2) \times (x \times x^2 \times x) \times (y \times y \times y^2) = 4x^4y^4 \)

4. For length \( = a \), breadth \( = 2b \), height \( = 3c \):
Volume \( = a \times 2b \times 3c = (1 \times 2 \times 3) \times a \times b \times c = 6abc \)
In simple words: To get the volume of a box, you simply multiply its three measurements: length, width, and height. Remember to multiply the numbers first, then combine the letters by adding their powers if they are the same.

Exam Tip: When calculating volume, ensure that all three dimensions (length, breadth, height) are expressed in the same units. Also, remember to add exponents for the same variable when multiplying terms, e.g., \( a \times a^2 \times a^4 = a^{1+2+4} = a^7 \).

Question 5. obtain the product of
1. \( xy, yz, zx \)
2. \( a, -a^2, a^3 \)
3. \( 2, 4y, 8y^2, 16y^3 \)
4. \( a, 2b, 3c, 6abc \)
5. \( m, -mn, mnp \)
Answer:
1. To find the product of \( xy, yz, zx \):
\( xy \times yz \times zx = (1 \times 1 \times 1) \times (x \times x) \times (y \times y) \times (z \times z) = x^2y^2z^2 \)

2. To find the product of \( a, -a^2, a^3 \):
\( a \times (-a^2) \times a^3 = [1 \times (-1) \times 1] \times (a \times a^2 \times a^3) = -a^6 \)

3. To find the product of \( 2, 4y, 8y^2, 16y^3 \):
\( 2 \times 4y \times 8y^2 \times 16y^3 = (2 \times 4 \times 8 \times 16) \times (y \times y^2 \times y^3) = 1024y^6 \)

4. To find the product of \( a, 2b, 3c, 6abc \):
\( a \times 2b \times 3c \times 6abc = (1 \times 2 \times 3 \times 6) \times (a \times a) \times (b \times b) \times (c \times c) = 36a^2b^2c^2 \)

5. To find the product of \( m, -mn, mnp \):
\( m \times (-mn) \times mnp = [1 \times (-1) \times 1] \times (m \times m \times m) \times (n \times n) \times p = -m^3n^2p \)
In simple words: When asked to find the product of several monomials, you multiply all the numerical parts together first. Then, for each variable, you combine all instances of that variable by adding their exponents. Be careful with negative signs!

Exam Tip: Pay close attention to the exponent of each variable. If a variable doesn't have an exponent written, it's understood to be 1. For example, \( y \) is \( y^1 \).