CBSE Class 8 Mathematics Algebraic Expressions And Identities Worksheet Set 07

Algebraic Expressions and Identities

Question 1. Using identity \((x - a)(x + a) = x^2 - a^2\) find \(6^2 - 5^2\).
Answer: Comparing \(6^2 - 5^2\) with the identity \(x^2 - a^2 = (x - a)(x + a)\), we get \(x = 6\) and \(a = 5\).
\(6^2 - 5^2 = (6 - 5)(6 + 5)\)
\(= (1)(11) = 11\)
In simple words: We used the difference of squares identity to evaluate the expression directly.

Exam Tip: Always identify \(x\) and \(a\) clearly before applying standard algebraic identities.

Question 2. Find the product of \((7x - 4y)\) and \((3x - 7y)\).
Answer: Product = \((7x - 4y)(3x - 7y)\)
\(= 7x(3x - 7y) - 4y(3x - 7y)\)
\(= 21x^2 - 49xy - 12xy + 28y^2\)
\(= 21x^2 - 61xy + 28y^2\)
In simple words: Multiply each term of the first binomial with every term of the second binomial and combine like terms.

Exam Tip: Be careful with negative signs when opening brackets during binomial multiplication.

Question 3. Using suitable identity find \((a + 3)(a + 2)\).
Answer: Using the identity \((x + a)(x + b) = x^2 + (a + b)x + ab\), where \(x = a\), \(a = 3\), and \(b = 2\):
\((a + 3)(a + 2) = a^2 + (3 + 2)a + (3)(2)\)
\(= a^2 + 5a + 6\)
In simple words: Apply the standard binomial product formula for terms with a common variable.

Exam Tip: Substitute the values directly into the formula to avoid calculation errors.

Question 4. Using identity \((a + b)^2 = a^2 + 2ab + b^2\) find the value of \(103^2\).
Answer: Express 103 as \((100 + 3)\).
\(103^2 = (100 + 3)^2\)
Here \(a = 100\) and \(b = 3\).
\(= (100)^2 + 2(100)(3) + (3)^2\)
\(= 10000 + 600 + 9\)
\(= 10609\)
In simple words: Split the number into a multiple of 10 and a small digit to use the square identity.

Exam Tip: Splitting numbers into powers of 10 makes squaring mental math much easier.

Question 5. Using identity \((a - b)^2 = a^2 - 2ab + b^2\) find the value of \(98^2\).
Answer: Express 98 as \((100 - 2)\).
\(98^2 = (100 - 2)^2\)
Here \(a = 100\) and \(b = 2\).
\(= (100)^2 - 2(100)(2) + (2)^2\)
\(= 10000 - 400 + 4\)
\(= 9604\)
In simple words: Write 98 as \(100 - 2\) and apply the binomial square formula.

Question 6. Using identity find \((2x + 3)^2\).
Answer: Using \((a + b)^2 = a^2 + 2ab + b^2\) with \(a = 2x\) and \(b = 3\):
\((2x + 3)^2 = (2x)^2 + 2(2x)(3) + (3)^2\)
\(= 4x^2 + 12x + 9\)
In simple words: Square the first term, add twice the product of both terms, and square the last term.

Exam Tip: Remember to square both the coefficient and the variable when evaluating \((2x)^2\).

Question 7. Subtract \(7x - 3x^2\) from \(4x + 8x^2\).
Answer: \((4x + 8x^2) - (7x - 3x^2)\)
\(= 4x + 8x^2 - 7x + 3x^2\)
\(= (8x^2 + 3x^2) + (4x - 7x)\)
\(= 11x^2 - 3x\)
In simple words: Group like terms together after distributing the negative sign.

Exam Tip: The phrase "subtract A from B" means writing it as \(B - A\).

Question 8. Using suitable identity find \((7x - 3y)^2\).
Answer: Using \((a - b)^2 = a^2 - 2ab + b^2\) with \(a = 7x\) and \(b = 3y\):
\((7x - 3y)^2 = (7x)^2 - 2(7x)(3y) + (3y)^2\)
\(= 49x^2 - 42xy + 9y^2\)
In simple words: Apply the square of difference identity directly.

Exam Tip: Pay close attention to the middle term sign in subtraction identities.

Question 9. Add \(4x^2 + 2xy - 4\) and \(7x^2 - 3xy + 4\).
Answer: \((4x^2 + 2xy - 4) + (7x^2 - 3xy + 4)\)
\(= (4x^2 + 7x^2) + (2xy - 3xy) + (-4 + 4)\)
\(= 11x^2 - xy + 0\)
\(= 11x^2 - xy\)
In simple words: Combine the coefficients of identical variable terms.

Exam Tip: Terms that cancel out completely need not be written in the final answer.

Question 10. Find the product of \(4x\), \(7x^2\), \(-2x\).
Answer: Product = \((4x) \cdot (7x^2) \cdot (-2x)\)
\(= (4 \cdot 7 \cdot -2) \cdot (x \cdot x^2 \cdot x)\)
\(= -56x^{1+2+1}\)
\(= -56x^4\)
In simple words: Multiply the numerical coefficients together and add the exponents of the same base variable.

Exam Tip: Use the law of exponents \(a^m \cdot a^n = a^{m+n}\) when multiplying variables.

Question 11. Find the product of \((x^2 - y^2)(2x + y)\).
Answer: \((x^2 - y^2)(2x + y)\)
\(= x^2(2x + y) - y^2(2x + y)\)
\(= 2x^3 + x^2y - 2xy^2 - y^3\)
In simple words: Distribute each term of the first binomial into the second binomial.

Question 12. Simplify: \((xy + yz)^2 - (xy - yz)^2\).
Answer: Method 1: Using identity \(a^2 - b^2 = (a - b)(a + b)\)
Let \(a = (xy + yz)\) and \(b = (xy - yz)\).
\([ (xy + yz) - (xy - yz) ] [ (xy + yz) + (xy - yz) ]\)
\(= [ xy + yz - xy + yz ] [ xy + yz + xy - yz ]\)
\(= [ 2yz ] [ 2xy ]\)
\(= 4xy^2z\)
In simple words: Use the difference of squares identity to simplify the expression efficiently.

Exam Tip: Recognizing standard identities saves time compared to expanding each term separately.

Question 13. Using identity find the product of \(\left(\frac{a}{2} + \frac{3b}{4}\right)\left(\frac{a}{2} + \frac{3b}{4}\right)\).
Answer: The expression can be written as \(\left(\frac{a}{2} + \frac{3b}{4}\right)^2\).
Using \((a + b)^2 = a^2 + 2ab + b^2\):
\(= \left(\frac{a}{2}\right)^2 + 2\left(\frac{a}{2}\right)\left(\frac{3b}{4}\right) + \left(\frac{3b}{4}\right)^2\)
\(= \frac{a^2}{4} + \frac{3ab}{4} + \frac{9b^2}{16}\)
In simple words: Apply the square of a sum identity with fractional terms.

Exam Tip: Square both the numerator and denominator of fractions carefully.

Question 14. Multiply: \((a^2 + 2c^2)(3a - 3c)\).
Answer: \((a^2 + 2c^2)(3a - 3c)\)
\(= a^2(3a - 3c) + 2c^2(3a - 3c)\)
\(= 3a^3 - 3a^2c + 6ac^2 - 6c^3\)
In simple words: Multiply each term in the first bracket by each term in the second bracket.

Exam Tip: Write variables in alphabetical order for clarity in final answers.

Question 15. Simplify: \((x + y)(2x - 3y + z) - (2x - 3y)z\).
Answer: Expand the first product:
\(= x(2x - 3y + z) + y(2x - 3y + z) - (2xz - 3yz)\)
\(= 2x^2 - 3xy + xz + 2xy - 3y^2 + yz - 2xz + 3yz\)
Group like terms:
\(= 2x^2 - 3y^2 + (-3xy + 2xy) + (xz - 2xz) + (yz + 3yz)\)
\(= 2x^2 - 3y^2 - xy - xz + 4yz\)
In simple words: Expand all brackets and collect terms with identical variables.

Exam Tip: Check every sign change carefully when removing parentheses.

Question 16. Subtract \(3x(x - 4y + 5z)\) from \(4x(2x - 3y + 10z)\).
Answer: First expression = \(4x(2x - 3y + 10z) = 8x^2 - 12xy + 40xz\)
Second expression = \(3x(x - 4y + 5z) = 3x^2 - 12xy + 15xz\)
Subtracting second from first:
\((8x^2 - 12xy + 40xz) - (3x^2 - 12xy + 15xz)\)
\(= 8x^2 - 12xy + 40xz - 3x^2 + 12xy - 15xz\)
\(= (8x^2 - 3x^2) + (-12xy + 12xy) + (40xz - 15xz)\)
\(= 5x^2 + 25xz\)
In simple words: Expand both expressions first, then subtract term by term.

Exam Tip: Like terms with opposite signs cancel out completely during subtraction.

Question 17. Simplify: \((x^2 - y^2)^2\).
Answer: Using \((a - b)^2 = a^2 - 2ab + b^2\) with \(a = x^2\) and \(b = y^2\):
\((x^2 - y^2)^2 = (x^2)^2 - 2(x^2)(y^2) + (y^2)^2\)
\(= x^4 - 2x^2y^2 + y^4\)
In simple words: Apply the square formula directly using exponent rules.

Exam Tip: Remember that \((x^2)^2 = x^4\) by multiplying exponents.

Question 18. Using suitable identity find the product of \(\left(\frac{2}{3}x - 5\right)\left(\frac{2}{3}x + 5\right)\).
Answer: Using \((x - a)(x + a) = x^2 - a^2\) with \(x = \frac{2}{3}x\) and \(a = 5\):
\(\left(\frac{2}{3}x - 5\right)\left(\frac{2}{3}x + 5\right) = \left(\frac{2}{3}x\right)^2 - (5)^2\)
\(= \frac{4}{9}x^2 - 25\)
In simple words: Use the difference of squares identity for identical terms with opposite signs.

Exam Tip: Square both numerator and denominator when squaring a fraction.

Question 19. Simplify \(3a(4a - 5) + 3\) and find its value for \(a = 3\).
Answer: Expansion:
\(3a(4a - 5) + 3 = 12a^2 - 15a + 3\)
Substitute \(a = 3\):
\(= 12(3)^2 - 15(3) + 3\)
\(= 12(9) - 45 + 3\)
\(= 108 - 45 + 3\)
\(= 66\)
In simple words: Expand the expression first and then substitute the given value of \(a\).

Exam Tip: Always simplify the algebraic expression before plugging in numerical values.

Question 20. Using suitable identity find \(\left(6x^2 - \frac{5}{3}\right)^2\).
Answer: Using \((a - b)^2 = a^2 - 2ab + b^2\) with \(a = 6x^2\) and \(b = \frac{5}{3}\):
\(\left(6x^2 - \frac{5}{3}\right)^2 = (6x^2)^2 - 2(6x^2)\left(\frac{5}{3}\right) + \left(\frac{5}{3}\right)^2\)
\(= 36x^4 - 2(2x^2)(5) + \frac{25}{9}\)
\(= 36x^4 - 20x^2 + \frac{25}{9}\)
In simple words: Expand using the square of difference identity and simplify fractions.

Question 21. Using identity \(a^2 - b^2 = (a + b)(a - b)\), find \((1.02)^2 - (0.98)^2\).
Answer: \((1.02)^2 - (0.98)^2 = (1.02 + 0.98)(1.02 - 0.98)\)
\(= (2.00)(0.04)\)
\(= 0.08\)
In simple words: Apply the difference of squares identity to decimal numbers.

Exam Tip: This method avoids complex decimal multiplication.

Question 22. Using \((x + a)(x + b) = x^2 + (a + b)x + ab\) find \(105 \times 107\).
Answer: Express as \((100 + 5)(100 + 7)\).
Here \(x = 100\), \(a = 5\), \(b = 7\).
\(= (100)^2 + (5 + 7)(100) + (5)(7)\)
\(= 10000 + (12)(100) + 35\)
\(= 10000 + 1200 + 35\)
\(= 11235\)
In simple words: Split numbers near a base of 100 and use the algebraic product identity.

Exam Tip: Choose base 100 for numbers slightly greater than 100.

Question 23. Using identity find the value of \((7.2)^2\).
Answer: \((7.2)^2 = (7 + 0.2)^2\)
Using \((a + b)^2 = a^2 + 2ab + b^2\):
\(= (7)^2 + 2(7)(0.2) + (0.2)^2\)
\(= 49 + 2.8 + 0.04\)
\(= 51.84\)
In simple words: Split 7.2 into integer and decimal parts before squaring.

Question 24. Using identity evaluate \(297 \times 303\).
Answer: Express as \((300 - 3)(300 + 3)\).
Using \((a - b)(a + b) = a^2 - b^2\):
\(= (300)^2 - (3)^2\)
\(= 90000 - 9\)
\(= 89991\)
In simple words: Express the numbers around 300 and apply the difference of squares identity.

Exam Tip: This is a classic application of the product of sum and difference identity.

Question 25. Using identity find the value of \((4.7)^2\).
Answer: \((4.7)^2 = (5 - 0.3)^2\)
Using \((a - b)^2 = a^2 - 2ab + b^2\):
\(= (5)^2 - 2(5)(0.3) + (0.3)^2\)
\(= 25 - 3 + 0.09\)
\(= 22 + 0.09\)
\(= 22.09\)
In simple words: Write 4.7 as \((5 - 0.3)\) and use the square of difference formula.

Exam Tip: Choosing numbers close to integers simplifies calculations.

Question 26. Simplify \(\left(\frac{5}{3}x + \frac{3}{4}y\right)^2 - \left(\frac{5}{3}x - \frac{3}{4}y\right)^2\) and also evaluate it when \(x = 2\) and \(y = -1\).
Answer: Using \(a^2 - b^2 = (a - b)(a + b)\):
\(= \left[\left(\frac{5}{3}x + \frac{3}{4}y\right) - \left(\frac{5}{3}x - \frac{3}{4}y\right)\right] \left[\left(\frac{5}{3}x + \frac{3}{4}y\right) + \left(\frac{5}{3}x - \frac{3}{4}y\right)\right]\)
\(= \left[\frac{5}{3}x + \frac{3}{4}y - \frac{5}{3}x + \frac{3}{4}y\right] \left[\frac{5}{3}x + \frac{3}{4}y + \frac{5}{3}x - \frac{3}{4}y\right]\)
\(= \left[\frac{6}{4}y\right]\left[\frac{10}{3}x\right]\)
\(= \left[\frac{3}{2}y\right]\left[\frac{10}{3}x\right]\)
\(= 5xy\)
Now evaluate for \(x = 2\) and \(y = -1\):
\(= 5(2)(-1)\)
\(= -10\)
In simple words: Simplify the expression using algebraic identities before substituting the values.

Exam Tip: Simplifying first reduces the complexity of arithmetic substitution.

Question 27. Simplify \((xy + yz)^2 - 2x^2y^2z\). Find the value when \(x = -1\), \(y = 1\) and \(z = 2\).
Answer: Expand \((xy + yz)^2\):
\(= (xy)^2 + 2(xy)(yz) + (yz)^2 - 2x^2y^2z\)
\(= x^2y^2 + 2xy^2z + y^2z^2 - 2x^2y^2z\)
Substitute \(x = -1\), \(y = 1\), \(z = 2\):
\(= (-1)^2(1)^2 + 2(-1)(1)^2(2) + (1)^2(2)^2 - 2(-1)^2(1)^2(2)\)
\(= (1)(1) + 2(-1)(1)(2) + (1)(4) - 2(1)(1)(2)\)
\(= 1 - 4 + 4 - 4\)
\(= -3\)
In simple words: Expand the square term, substitute the given values, and compute step by step.

Exam Tip: Take extra care with negative signs during exponentiation and multiplication.

Question 28. Simplify: \((1.5x - 4y)(1.5x + 4y + 3) - 4.5x + 12y\).
Answer: Expand the product:
\(= 1.5x(1.5x + 4y + 3) - 4y(1.5x + 4y + 3) - 4.5x + 12y\)
\(= 2.25x^2 + 6xy + 4.5x - 6xy - 16y^2 - 12y - 4.5x + 12y\)
Group like terms:
\(= 2.25x^2 - 16y^2 + (6xy - 6xy) + (4.5x - 4.5x) + (-12y + 12y)\)
\(= 2.25x^2 - 16y^2\)
In simple words: Multiply out the terms and cancel opposite terms.

Exam Tip: Many terms cancel out completely in simplification problems.

Most Important Questions

Question 1. What are algebraic expressions?
Answer: An algebraic expression is a combination of constants and variables connected by mathematical operations such as addition, subtraction, multiplication, and division.
In simple words: Expressions combine numbers and letters using math operations.

Exam Tip: Remember that expressions do not have an equals sign.

Question 2. Expressions consists of _____________ & _______________ .
Answer: Answer: Variables & Constants
In simple words: Expressions are built using letters (variables) and fixed numbers (constants).

Exam Tip: Constants have fixed values while variables can change.

Question 3. T/F. The value of an expression changes with the value chosen for the variables it contains.
Answer: True
In simple words: Changing the variable's value changes the result of the expression.

Exam Tip: Variables determine the output value of an algebraic expression.

Question 4. When numbers/literals are added or subtracted, they are called _________ .
Answer: Answer: Terms
In simple words: Parts of an expression separated by plus or minus signs are called terms.

Exam Tip: Terms form the building blocks of polynomials.

Question 5. When numbers/literals are multiplied, they are called ________.
Answer: Answer: Factors
In simple words: Numbers or variables multiplied together to form a term are its factors.

Exam Tip: Factors make up a term through multiplication.

Question 6. The terms in the expression \(4ab + 5a(b + c)\) are:
Answer: Expanding \(5a(b + c) = 5ab + 5ac\), the expression is \(4ab + 5ab + 5ac = 9ab + 5ac\). The terms are \(9ab\) and \(5ac\).
In simple words: Simplify the expression first, then list its individual terms.

Exam Tip: Always expand brackets before identifying final terms.

Question 7. The factors in the term \(5a(b + c)\) are :
Answer: The factors are \(5\), \(a\), and \((b + c)\).
In simple words: The components multiplied together in the term are its factors.

Exam Tip: Parenthesized expressions count as single factors.

Question 8. A monomial is an expression in which ____________________ .
Answer: Answer: there is only one term
In simple words: Monomials consist of a single algebraic term.

Exam Tip: Example: \(5x^2\) is a monomial.

Question 9. A binomial is an expression in which ___________________ .
Answer: Answer: there are exactly two terms
In simple words: Binomials contain two unlike terms connected by plus or minus.

Exam Tip: Example: \(x + y\) is a binomial.

Question 10. While multiplying two monomials, Coefficient of product = _________ X ___________.
Answer: Answer: Coefficient of first monomial X Coefficient of second monomial
In simple words: Multiply numerical coefficients together to get the new coefficient.

Exam Tip: Coefficients multiply separately from variables.

Question 11. Identify the terms, their coefficients for the expression: \(0.75x + 0.44y + 1.56zx\)
Answer: Terms: \(0.75x\), \(0.44y\), \(1.56zx\).
Coefficients: 0.75 for \(x\), 0.44 for \(y\), 1.56 for \(zx\).
In simple words: Identify each term and its numerical multiplier.

Exam Tip: The coefficient is the numerical factor of a term.

Question 12. Classify the following as binomials and trinomials: \(2a + 3b\), \(2x + 3y - 5\), \(a + 4\), \(12x + 13y + 17z\)
Answer: Binomials: \(2a + 3b\), \(a + 4\)
Trinomials: \(2x + 3y - 5\), \(12x + 13y + 17z$
In simple words: Count the terms: two terms = binomial, three terms = trinomial.

Exam Tip: Simplify expressions before classifying them.

Question 13. What are the polynomials? Give an example.
Answer: An expression with one or more terms having non-negative integer exponents for variables is called a polynomial. Example: \(3x^2 + 2x - 5\).
In simple words: Polynomials are algebraic expressions with multiple terms.

Exam Tip: Variables cannot have negative or fractional exponents in polynomials.

Question 14. What are like and unlike terms?
Answer: Like terms have the same variables with the same exponents. Unlike terms have different variables or different exponents.
In simple words: Like terms match perfectly in their variables and powers.

Question 15. Classify as like and unlike terms: \(2abc\) and \(bac\), \(x^2y^2z\) and \(y^2zx^2\), \(7x\) and \(3y\), \(xy + z\) and \(xyz\)
Answer: Like terms: \(2abc\) and \(bac\), \(x^2y^2z\) and \(y^2zx^2\).
Unlike terms: \(7x\) and \(3y\), \(xy + z\) and \(xyz\).
In simple words: Check if variable parts match exactly.

Exam Tip: Multiplication order of variables does not affect like terms (\(abc = bac\)).

Question 16. Add: \(2p^2q^2 - 3pq + 4\) & \(5 + 7pq - 3p^2q^2\)
Answer: \((2p^2q^2 - 3pq + 4) + (5 + 7pq - 3p^2q^2)\)
\(= (2p^2q^2 - 3p^2q^2) + (-3pq + 7pq) + (4 + 5)\)
\(= -p^2q^2 + 4pq + 9$
In simple words: Group and add coefficients of like terms.

Exam Tip: Combine all constant terms together.

Question 17. Add: \(-8x^2 + 2y^2z - 11\) and \(-2x^2 + 9y^2z - 1\)
Answer: \((-8x^2 + 2y^2z - 11) + (-2x^2 + 9y^2z - 1)\)
\(= (-8x^2 - 2x^2) + (2y^2z + 9y^2z) + (-11 - 1)\)
\(= -10x^2 + 11y^2z - 12$
In simple words: Add coefficients of matching variable terms.

Question 18. Subtract \(4p^2q + 5pq + 5pq^2 - 8p + 7q - 10\) from \(18 - 3p - 11q + 5pq - 2pq^2 + 5p^2q\).
Answer: \((18 - 3p - 11q + 5pq - 2pq^2 + 5p^2q) - (4p^2q + 5pq + 5pq^2 - 8p + 7q - 10)\)
\(= 18 - 3p - 11q + 5pq - 2pq^2 + 5p^2q - 4p^2q - 5pq - 5pq^2 + 8p - 7q + 10$
Grouping like terms:
\(= (5p^2q - 4p^2q) + (-2pq^2 - 5pq^2) + (5pq - 5pq) + (-3p + 8p) + (-11q - 7q) + (18 + 10)\)
\(= p^2q - 7pq^2 + 5p - 18q + 28$
In simple words: Subtract each corresponding term after distributing the negative sign.

Exam Tip: Write terms in a structured order to avoid missing any part.

Question 19. Find the product of: \(2z\), \(4y\), \(2y^2\) & \(6xyz\)
Answer: Product = \((2z)(4y)(2y^2)(6xyz)\)
\(= (2 \times 4 \times 2 \times 6)(z \cdot y \cdot y^2 \cdot x \cdot y \cdot z)\)
\(= 96 x y^{1+2+1} z^{1+1}\)
\(= 96 x y^4 z^2$
In simple words: Multiply numbers together and add exponents of identical variables.

Exam Tip: Count all occurrences of each variable carefully.

Question 20. Find: a) \(13mn \times 13np\) b) \(-4xy \times -7x^2y\)
Answer:
a) \(13mn \times 13np = (13 \times 13)(m \cdot n \cdot n \cdot p) = 169mn^2p$
b) \(-4xy \times -7x^2y = (-4 \times -7)(x \cdot x^2 \cdot y \cdot y) = 28x^3y^2$
In simple words: Multiply numerical parts and combine variable powers.

Exam Tip: Negative times negative gives a positive result.

Question 21. State the distributive property.
Answer: The distributive property states that for any numbers \(a\), \(b\), and \(c\): \(a(b + c) = ab + ac\).
In simple words: Multiplying a number by a sum is the same as multiplying by each addend separately.

Exam Tip: This property is fundamental for expanding algebraic brackets.

Question 22. T/F: \(n(4 + m) = 4n + nm\)
Answer: True
In simple words: Distributing \(n\) into the bracket yields \(4n + nm\).

Exam Tip: Multiplication is commutative, so \(nm = mn\).

Question 23. T/F: \(p(9 - p) = 9p - 2p\)
Answer: False (Correct expression is \(9p - p^2\)).
In simple words: Multiplying \(p\) by \(p\) gives \(p^2\), not \(2p\).

Exam Tip: Variable multiplication adds exponents; addition adds coefficients.

Question 24. Whenever we multiply a binomial by a binomial, we get _____terms in the product.
Answer: Answer: 4 (before combining like terms)
In simple words: Each of the two terms multiplies both terms, giving four initial products.

Exam Tip: These four terms may reduce to fewer terms if like terms combine.

Question 25. Whenever we multiply a binomial by a trinomial, we get ____terms in the product.
Answer: Answer: 6 (before combining like terms)
In simple words: Two terms multiplied by three terms produce six initial terms.

Question 26. Find: \(a^2(2ab - 5c)\)
Answer: \(= a^2(2ab) - a^2(5c)\)
\(= 2a^3b - 5a^2c$
In simple words: Distribute \(a^2\) to both terms inside the bracket.

Exam Tip: Add exponents for variable \(a\): \(a^2 \cdot a = a^3\).

Question 27. Simplify \(x(x - 3) + 2\) and evaluate for \(x = 2\).
Answer: Expansion:
\(x^2 - 3x + 2$
Substitute \(x = 2\):
\(= (2)^2 - 3(2) + 2$
\(= 4 - 6 + 2$
\(= 0$
In simple words: Expand the expression first, then substitute \(x = 2\).

Exam Tip: Evaluation becomes simpler after algebraic expansion.

Question 28. Find: \(\left(-\frac{7}{3}pq\right) \times \left(\frac{6}{5}pqr\right)\)
Answer: \(= \left(-\frac{7}{3} \times \frac{6}{5}\right) (pq \cdot pqr)\)
\(= \left(-\frac{42}{15}\right) p^{1+1} q^{1+1} r$
\(= -\frac{14}{5} p^2 q^2 r$
In simple words: Multiply numerical fractions and add matching variable powers.

Exam Tip: Simplify numerical fractions to lowest terms.

Question 29. Find: \((3x^6) \times (6x^{12}) \times (9x^{18})\)
Answer: \(= (3 \times 6 \times 9)(x^6 \cdot x^{12} \cdot x^{18})\)
\(= 162 x^{6 + 12 + 18}\)
\(= 162 x^{36}$
In simple words: Multiply coefficients and add all exponents of \(x\).

Question 30. Find using distributive property: \(125 \times 42\)
Answer: \(= 125 \times (40 + 2)\)
\(= (125 \times 40) + (125 \times 2)\)
\(= 5000 + 250$
\(= 5250$
In simple words: Split 42 into \((40 + 2)\) and distribute 125.

Exam Tip: Distributive property simplifies mental multiplication.

Question 31. Find: \(ab(a^2 + bc + c^2)\)
Answer: \(= ab(a^2) + ab(bc) + ab(c^2)\)
\(= a^3b + ab^2c + abc^2$
In simple words: Multiply \(ab\) across each term inside the parenthesis.

Exam Tip: Combine variables correctly during term multiplication.

Question 32. Find: \(5m(m^2 + m + 1)\) and evaluate for \(m = 1\).
Answer: Expansion:
\(= 5m(m^2) + 5m(m) + 5m(1)\)
\(= 5m^3 + 5m^2 + 5m$
Substitute \(m = 1\):
\(= 5(1)^3 + 5(1)^2 + 5(1)\)
\(= 5 + 5 + 5$
\(= 15$
In simple words: Expand the expression and substitute \(m = 1\).

Exam Tip: Powers of 1 always equal 1.

Question 33. Multiply: \((2.5a - 0.5b)\) and \((2.5a + 0.5b + c)\)
Answer: \(= 2.5a(2.5a + 0.5b + c) - 0.5b(2.5a + 0.5b + c)\)
\(= 6.25a^2 + 1.25ab + 2.5ac - 1.25ab - 0.25b^2 - 0.5bc$
\(= 6.25a^2 + 2.5ac - 0.25b^2 - 0.5bc$
In simple words: Distribute both terms of the binomial into the trinomial and combine like terms.

Exam Tip: Notice that terms like \(1.25ab\) and \(-1.25ab\) cancel out.

Question 34. Multiply: \((ab + 5)(a + c^2)(b + 6)\)
Answer: First multiply \((ab + 5)(b + 6) = ab(b + 6) + 5(b + 6) = ab^2 + 6ab + 5b + 30\).
Now multiply by \((a + c^2)\):
\(= (ab^2 + 6ab + 5b + 30)(a + c^2)\)
\(= a^2b^2 + ab^2c^2 + 6a^2b + 6abc^2 + 5ab + 5bc^2 + 30a + 30c^2$
In simple words: Multiply the binomials in pairs sequentially.

Exam Tip: Multiplying three binomials requires two successive multiplications.

Question 35. Simplify: \((m + n)(3m + n) + (m + 2n)(m - n)\)
Answer: Expand first product: \((m + n)(3m + n) = 3m^2 + mn + 3mn + n^2 = 3m^2 + 4mn + n^2\).
Expand second product: \((m + 2n)(m - n) = m^2 - mn + 2mn - 2n^2 = m^2 + mn - 2n^2$
Add both results:
\(= (3m^2 + 4mn + n^2) + (m^2 + mn - 2n^2)\)
\(= (3m^2 + m^2) + (4mn + mn) + (n^2 - 2n^2)\)
\(= 4m^2 + 5mn - n^2$
In simple words: Expand each product separately and then combine like terms.

Exam Tip: Keep brackets around expanded products until final addition.

Question 36. Simplify: \((3.5e - 4.5f)(1.5e + 4f + ef) - 4.5e + 10f\)
Answer: Expand the product:
\(3.5e(1.5e + 4f + ef) - 4.5f(1.5e + 4f + ef) - 4.5e + 10f$
\(= 5.25e^2 + 14ef + 3.5e^2f - 6.75ef - 18f^2 - 4.5ef^2 - 4.5e + 10f$
Group like terms:
\(= 5.25e^2 - 18f^2 + 3.5e^2f - 4.5ef^2 + (14ef - 6.75ef) - 4.5e + 10f$
\(= 5.25e^2 - 18f^2 + 3.5e^2f - 4.5ef^2 + 7.25ef - 4.5e + 10f$
In simple words: Expand carefully and collect all like terms.

Exam Tip: Double check decimal multiplication steps.

Question 37. Simplify: \((3.5e - 4.5f)(1.5e + 4f + ef) - 4.5e + 10f\)
Answer: (Identical to Question 36)
\(= 5.25e^2 - 18f^2 + 3.5e^2f - 4.5ef^2 + 7.25ef - 4.5e + 10f$
In simple words: Expand and simplify all terms as shown in Question 36.

Exam Tip: Repeated questions should be verified for consistency.

Question 38. What is an identity.
Answer: An identity is an equality that is true for all values of the variables in it.
In simple words: An equation that holds true for every possible variable value is an identity.

Exam Tip: Identities differ from equations, which are true only for specific values.

Question 39. True/False. An equation, which is true for only certain values of the variable in it, is not an identity.
Answer: True
In simple words: Only equations true for all values qualify as identities.

Exam Tip: Conditional equations are not identities.

Question 40. \((x + a)(x + b) =\)________________
Answer: Answer: \(x^2 + (a + b)x + ab$
In simple words: Standard product identity for binomials with a common term.

Exam Tip: Memorize this identity for quadratic expansions.

Question 41. \((a - b)^2 =\)_______________
Answer: Answer: \(a^2 - 2ab + b^2$
In simple words: Square of difference identity formula.

Exam Tip: The middle term is negative in difference squares.

Question 42. \((x - a)(x + a) =\)___________
Answer: Answer: \(x^2 - a^2$
In simple words: Difference of two squares identity.

Exam Tip: Useful for quickly multiplying terms with alternating signs.

Question 43. If \(x + \frac{1}{x} = 6\), find \(x^2 + \frac{1}{x^2}\).
Answer: Squaring both sides of \(x + \frac{1}{x} = 6\):
\(\left(x + \frac{1}{x}\right)^2 = 6^2$
\(x^2 + 2(x)\left(\frac{1}{x}\right) + \frac{1}{x^2} = 36$
\(x^2 + 2 + \frac{1}{x^2} = 36$
\(x^2 + \frac{1}{x^2} = 36 - 2$
\(x^2 + \frac{1}{x^2} = 34$
In simple words: Square the given equation to find the sum of reciprocal squares.

Exam Tip: This is a standard algebraic manipulation technique.

Question 44. If \(x + y = 12\) and \(xy = 32\). Find the value of \(x^2 + y^2\).
Answer: Using \((x + y)^2 = x^2 + 2xy + y^2\):
\((12)^2 = x^2 + 2(32) + y^2$
\(144 = x^2 + 64 + y^2$
\(x^2 + y^2 = 144 - 64$
\(x^2 + y^2 = 80$
In simple words: Substitute known values into the square of sum identity.

Exam Tip: Isolate the required term by shifting constants.

Question 45. Find using identities: \(106^2\)
Answer: \((100 + 6)^2 = (100)^2 + 2(100)(6) + (6)^2$
\(= 10000 + 1200 + 36$
\(= 11236$
In simple words: Write 106 as \((100 + 6)\) and apply the square formula.

Exam Tip: Using base 100 simplifies mental calculation.

Question 46. Find using identities: \((4.8)^2\)
Answer: \((5 - 0.2)^2 = (5)^2 - 2(5)(0.2) + (0.2)^2$
\(= 25 - 2 + 0.04$
\(= 23.04$
In simple words: Express 4.8 as \((5 - 0.2)\) and use the subtraction identity.

Exam Tip: Subtracting from a whole number is easier with decimals.

Question 47. Find using identities: \((-p + q)(-p + q)\)
Answer: \((-p + q)^2 = (-p)^2 - 2(-p)(q) + (q)^2$
\(= p^2 + 2pq + q^2$
In simple words: Square the binomial terms using the sum formula.

Exam Tip: \((-p)^2\) evaluates to positive \(p^2\).

Question 48. Find: \((2x + 5y)(2x + 3y)\)
Answer: Using \((x + a)(x + b) = x^2 + (a + b)x + ab\) with \(x = 2x\), \(a = 5y\), \(b = 3y\):
\(= (2x)^2 + (5y + 3y)(2x) + (5y)(3y)\)
\(= 4x^2 + (8y)(2x) + 15y^2$
\(= 4x^2 + 16xy + 15y^2$
In simple words: Apply the standard binomial product identity.

Question 49. Find: \((2x - y)(2x + y)(4x^2 + y^2)\)
Answer: First multiply \((2x - y)(2x + y) = (2x)^2 - (y)^2 = 4x^2 - y^2\).
Now multiply by \((4x^2 + y^2)\):
\(= (4x^2 - y^2)(4x^2 + y^2)\)
\(= (4x^2)^2 - (y^2)^2$
\(= 16x^4 - y^4$
In simple words: Apply the difference of squares identity twice.

Exam Tip: Spotting difference of squares simplifies multi-term products.

Question 50. Multiply: \([p^2 + (qr)^2][p^2 - (qr)^2]\) and evaluate for \(p = 1, q = 2, r = 3\).
Answer: Using \((a - b)(a + b) = a^2 - b^2\):
\([p^2 + (qr)^2][p^2 - (qr)^2] = (p^2)^2 - (qr)^2^2$
\(= p^4 - q^2r^2$
Substitute \(p = 1, q = 2, r = 3\):
\(= (1)^4 - (2)^2(3)^2$
\(= 1 - (4)(9)$
\(= 1 - 36$
\(= -35$
In simple words: Simplify using the identity before substituting the values.

Exam Tip: Algebraic simplification makes numerical evaluation much faster.

Question 51. Derive the identity: \((x + a)(x + b) = x^2 + (a + b)x + ab\)
Answer: \((x + a)(x + b) = x(x + b) + a(x + b)\)
\(= x \cdot x + x \cdot b + a \cdot x + a \cdot b$
\(= x^2 + bx + ax + ab$
\(= x^2 + (a + b)x + ab$
In simple words: Expand the binomials using distribution and combine like terms.

Exam Tip: Showing intermediate distribution steps ensures full marks in derivations.

Question 52. Find the product: \(\left(a - \frac{b}{2} - 1\right)\left(a + \frac{b}{2} + 1\right)\)
Answer: Rewrite as: \(\left[a - \left(\frac{b}{2} + 1\right)\right]\left[a + \left(\frac{b}{2} + 1\right)\right]\)
Using \((x - y)(x + y) = x^2 - y^2$
\(= a^2 - \left(\frac{b}{2} + 1\right)^2$
Expand \(\left(\frac{b}{2} + 1\right)^2 = \left(\frac{b}{2}\right)^2 + 2\left(\frac{b}{2}\right)(1) + (1)^2 = \frac{b^2}{4} + b + 1$
\(= a^2 - \left(\frac{b^2}{4} + b + 1\right)\)
\(= a^2 - \frac{b^2}{4} - b - 1$
In simple words: Group terms to form a difference of squares pattern.

Exam Tip: Grouping terms strategically makes complex products manageable.