RBSE Solutions Class 8 Maths Chapter 9 Algebraic Expressions Exercise 9.1

Rajasthan Board RBSE Class 8 Maths Chapter 9 Algebraic Expressions Ex 9.1

Question 1. Find the product of the following pairs of monomials.
(i) 3 x 5x
(ii) – 5p, - 2q
(iii) 7l², – 3n²
(iv) 6m, 3n
(v) - 5x², – 2x
Answer:
(i) To find the product of 3 and 5x, we multiply the numerical parts and then include the variable. So, \( 3 \times 5x = (3 \times 5) \times x = 15x \). This is a straightforward multiplication of a constant and a monomial.
(ii) To find the product of -5p and -2q, multiply the numerical coefficients and then the variable terms. \( (-5p) \times (-2q) = (-5) \times (-2) \times p \times q = 10pq \). Remember that multiplying two negative numbers results in a positive number.
(iii) For \( 7l^2 \) and \( -3n^2 \), multiply the numerical coefficients and then combine the variable terms. \( (7l^2) \times (-3n^2) = (7 \times -3) \times (l^2 \times n^2) = -21l^2n^2 \). Different variables with exponents are simply written next to each other.
(iv) For 6m and 3n, multiply the numerical parts and then combine the different variables. \( (6m) \times (3n) = (6 \times 3) \times (m \times n) = 18mn \). This is a simple product of two monomials with distinct variables.
(v) For \( -5x^2 \) and \( -2x \), multiply the numerical coefficients and then combine the 'x' terms by adding their exponents. \( (-5x^2) \times (-2x) = (-5) \times (-2) \times x^2 \times x = 10x^{(2+1)} = 10x^3 \). When variables are multiplied, their powers add up.
In simple words: For each part, first multiply the numbers together, keeping track of any negative signs. Then, write down the variables. If the same variable appears more than once, add their small numbers (exponents).

🎯 Exam Tip: Always pay close attention to the signs of the coefficients and remember the rules for multiplying exponents (add them for the same base). This helps prevent common errors.

Question 3. Multiply the following monomials
(i) xy, x²y, xy, x
(ii) m, n, mn, m³n, mn³
(iii) kl, lm, km, klm
Answer:
(i) To find the product of xy, \( x^2y \), xy, and x, we multiply all the 'x' terms and all the 'y' terms separately. For 'x' terms: \( x^1 \times x^2 \times x^1 \times x^1 = x^{(1+2+1+1)} = x^5 \). For 'y' terms: \( y^1 \times y^1 \times y^1 = y^{(1+1+1)} = y^3 \). So, the complete product is \( x^5y^3 \). This method ensures all variable instances are accounted for.
(ii) To find the product of m, n, mn, \( m^3n \), and \( mn^3 \), we multiply all the 'm' terms and all the 'n' terms by adding their exponents. For 'm' terms: \( m^1 \times m^1 \times m^3 \times m^1 = m^{(1+1+3+1)} = m^6 \). For 'n' terms: \( n^1 \times n^1 \times n^1 \times n^3 = n^{(1+1+1+3)} = n^6 \). Therefore, the total product is \( m^6n^6 \). This process applies the exponent rule for multiplication across multiple monomials.
(iii) To find the product of kl, lm, km, and klm, we group and multiply each variable type ('k', 'l', 'm') by adding their exponents. For 'k' terms: \( k^1 \times k^1 \times k^1 = k^{(1+1+1)} = k^3 \). For 'l' terms: \( l^1 \times l^1 \times l^1 = l^{(1+1+1)} = l^3 \). For 'm' terms: \( m^1 \times m^1 \times m^1 = m^{(1+1+1)} = m^3 \). The final product is \( k^3l^3m^3 \). When multiplying multiple variables, combine them.
In simple words: For each set of monomials, find all occurrences of each variable (like 'x', 'y', 'm', 'n', 'k', 'l'). Then, add the small power numbers (exponents) for each variable to get its total power in the final answer. Combine these new powers for all variables.

🎯 Exam Tip: When multiplying multiple monomials, a good practice is to write out all the variables, count their individual powers (remember a variable without a power shown has a power of 1), and then sum these powers for each unique variable to form the final product.

Question 4. Find the simple interest (SI) using the formula \( SI = \frac { PTR }{100} \), if Principal (P) \( = 4x^2 \), Time (T) \( = 5x \) and Rate of Interest (R) \( = 5y \).
Answer: We are given the principal (P), time (T), and rate of interest (R) for calculating simple interest (SI).
Principal (P) \( = 4x^2 \)
Time (T) \( = 5x \)
Rate of Interest (R) \( = 5y \)
The formula for simple interest is: \( SI = \frac { P \times T \times R }{100} \)
Now, substitute the given values into the formula:
\( SI = \frac { (4x^2) \times (5x) \times (5y) }{ 100 } \)
First, multiply the numerical coefficients in the numerator:
\( 4 \times 5 \times 5 = 100 \)
Next, multiply the variable parts: \( x^2 \times x \times y = x^{(2+1)} \times y = x^3y \)
Combine these results for the numerator:
\( SI = \frac { 100 \times x^3y }{ 100 } \)
Now, cancel out the common factor of 100 from the numerator and the denominator:
\( SI = x^3y \)
Thus, the simple interest is \( x^3y \). Substituting the values into the formula and simplifying gives the final expression for the interest.
In simple words: We put the numbers and letters for Principal, Time, and Rate into the Simple Interest formula. We multiply all the numbers on top to get 100, and all the 'x's to get \( x^3 \), keeping 'y' as it is. Then, we divide the whole thing by 100. This leaves us with \( x^3y \) as the final simple interest.

🎯 Exam Tip: Always write down the given values and the formula first. Then, substitute carefully, paying attention to the exponents of variables. Simplify by multiplying numbers and combining variables, then cancel any common factors.