Identities

Q1: Multiple Choice Type:

i. \(\left(x+2y\right)^2+\left(x-2y\right)^2\) is equal to:

Step 1: Expand each square using \((p+q)^2 = p^2 + 2pq + q^2\) and \((p-q)^2 = p^2 – 2pq + q^2\)
\((x+2y)^2 = x^2 + 4xy + 4y^2\)
\((x-2y)^2 = x^2 – 4xy + 4y^2\)
Step 2: Add the two expressions
\((x+2y)^2 + (x-2y)^2 = (x^2 + 4xy + 4y^2) + (x^2 – 4xy + 4y^2) = 2x^2 + 8y^2\)
Answer: c. \(2x^2 + 8y^2\)

ii. \(\left(a+b\right)\left(a-b\right)+\left(b-c\right)\left(b+c\right)+\left(c+a\right)\left(c-a\right)\) is equal to:

Step 1: Use the identity \((p+q)(p-q) = p^2 – q^2\)
\((a+b)(a-b) = a^2 – b^2\)
\((b-c)(b+c) = b^2 – c^2\)
\((c+a)(c-a) = c^2 – a^2\)
Step 2: Add all terms
\((a^2 – b^2) + (b^2 – c^2) + (c^2 – a^2) = 0\)
Answer: c. 0

iii. \(\left(3x-4y\right)^2-\left(3x+4y\right)^2\) is equal to:

Step 1: Use the identity \((p-q)^2 – (p+q)^2 = -4pq\)
\((3x-4y)^2 – (3x+4y)^2 = -4 \cdot (3x) \cdot (4y) = -48xy\)
Answer: c. -48xy

iv. The value of \(\left(0.8\right)^2-0.32+\left(0.2\right)^2\) is equal to:

Step 1: Calculate squares
\((0.8)^2 = 0.64\), \((0.2)^2 = 0.04\)
Step 2: Substitute and simplify
\(0.64 – 0.32 + 0.04 = 0.36\)
Answer: c. 0.36

v. The value of \(\left(a-b-c\right)\left(a-b+c\right)\) is:

Step 1: Use identity \((p-q)(p+q) = p^2 – q^2\) with \(p = a-b\) and \(q = c\)
\((a-b-c)(a-b+c) = (a-b)^2 – c^2\)
Step 2: Expand \((a-b)^2 – c^2\)
\((a-b)^2 – c^2 = a^2 – 2ab + b^2 – c^2 = a^2 + b^2 – c^2 – 2ab\)
Answer: b. \(a^2 + b^2 – c^2 – 2ab\)

Q2: Use direct method evaluate the following product:

i. \((a-8)(a+2)\)

Step 1: Apply distributive property: \((p+q)(r+s) = pr + ps + qr + qs\)
\((a-8)(a+2) = a\cdot a + a\cdot 2 + (-8)\cdot a + (-8)\cdot 2\)
Step 2: Simplify each term
\(a^2 + 2a – 8a – 16\)
Step 3: Combine like terms
\(a^2 – 6a – 16\)
Answer:\(a^2 – 6a – 16\)

ii. \((b-3)(b-5)\)

Step 1: Apply distributive property
\((b-3)(b-5) = b\cdot b + b\cdot (-5) + (-3)\cdot b + (-3)\cdot (-5)\)
Step 2: Simplify each term
\(b^2 – 5b – 3b + 15\)
Step 3: Combine like terms
\(b^2 – 8b + 15\)
Answer:\(b^2 – 8b + 15\)

iii. \((3x-2y)(2x+y)\)

Step 1: Apply distributive property
\((3x-2y)(2x+y) = 3x\cdot 2x + 3x\cdot y + (-2y)\cdot 2x + (-2y)\cdot y\)
Step 2: Simplify each term
\(6x^2 + 3xy – 4xy – 2y^2\)
Step 3: Combine like terms
\(6x^2 – xy – 2y^2\)
Answer:\(6x^2 – xy – 2y^2\)

iv. \((5a+16)(3a-7)\)

Step 1: Apply distributive property
\((5a+16)(3a-7) = 5a\cdot 3a + 5a\cdot (-7) + 16\cdot 3a + 16\cdot (-7)\)
Step 2: Simplify each term
\(15a^2 – 35a + 48a – 112\)
Step 3: Combine like terms
\(15a^2 + 13a – 112\)
Answer:\(15a^2 + 13a – 112\)

v. \((8-b)(3+b)\)

Step 1: Apply distributive property
\((8-b)(3+b) = 8\cdot 3 + 8\cdot b + (-b)\cdot 3 + (-b)\cdot b\)
Step 2: Simplify each term
\(24 + 8b – 3b – b^2\)
Step 3: Combine like terms
\(-b^2 + 5b + 24\)
Answer:\(-b^2 + 5b + 24\)

Q3: Evaluate:

i. \((2a+3)(2a-3)\)

Step 1: Apply the identity \((p+q)(p-q) = p^2 – q^2\)
Here, \(p = 2a\), \(q = 3\)
\((2a+3)(2a-3) = (2a)^2 – 3^2\)
Step 2: Simplify
\(4a^2 – 9\)
Answer:\(4a^2 – 9\)

ii. \((xy+4)(xy-4)\)

Step 1: Apply \((p+q)(p-q) = p^2 – q^2\), \(p = xy, q = 4\)
\((xy+4)(xy-4) = (xy)^2 – 16 = x^2y^2 – 16\)
Answer:\(x^2y^2 – 16\)

iii. \((ab + x^2)(ab – x^2)\)

Step 1: Apply identity with \(p = ab, q = x^2\)
\((ab + x^2)(ab – x^2) = (ab)^2 – (x^2)^2\)
Step 2: Simplify
\(a^2b^2 – x^4\)
Answer:\(a^2b^2 – x^4\)

iv. \((3x^2 + 5y^2)(3x^2 – 5y^2)\)

Step 1: Apply identity \((p+q)(p-q) = p^2 – q^2\), \(p = 3x^2, q = 5y^2\)
\((3x^2 + 5y^2)(3x^2 – 5y^2) = (3x^2)^2 – (5y^2)^2\)
Step 2: Simplify
\(9x^4 – 25y^4\)
Answer:\(9x^4 – 25y^4\)

v. \(\left(z – \frac{2}{3}\right)\left(z + \frac{2}{3}\right)\)

Step 1: Apply \((p+q)(p-q) = p^2 – q^2\), \(p = z, q = \frac{2}{3}\)
\(\left(z – \frac{2}{3}\right)\left(z + \frac{2}{3}\right) = z^2 – \left(\frac{2}{3}\right)^2\)
Step 2: Simplify
\(z^2 – \frac{4}{9}\)
Answer:\(z^2 – \frac{4}{9}\)

vi. \(\left(\frac{3}{5}a + \frac{1}{2}\right)\left(\frac{3}{5}a – \frac{1}{2}\right)\)

Step 1: Use \((p+q)(p-q) = p^2 – q^2\), \(p = \frac{3}{5}a, q = \frac{1}{2}\)
\(\left(\frac{3}{5}a + \frac{1}{2}\right)\left(\frac{3}{5}a – \frac{1}{2}\right) = \left(\frac{3}{5}a\right)^2 – \left(\frac{1}{2}\right)^2\)
Step 2: Simplify
\(\frac{9}{25}a^2 – \frac{1}{4}\)
Answer:\( \frac{9}{25} a^2 – \frac{1}{4}\)

vii. \((0.5 – 2a)(0.5 + 2a)\)

Step 1: Apply \((p+q)(p-q) = p^2 – q^2\), \(p = 0.5, q = 2a\)
\((0.5 – 2a)(0.5 + 2a) = (0.5)^2 – (2a)^2\)
Step 2: Simplify
\(0.25 – 4a^2\)
Answer:\(0.25 – 4a^2\)

viii. \(\left(\frac{a}{2} – \frac{b}{3}\right)\left(\frac{a}{2} + \frac{b}{3}\right)\)

Step 1: Apply identity \((p-q)(p+q) = p^2 – q^2\), \(p = \frac{a}{2}, q = \frac{b}{3}\)
\(\left(\frac{a}{2} – \frac{b}{3}\right)\left(\frac{a}{2} + \frac{b}{3}\right) = \left(\frac{a}{2}\right)^2 – \left(\frac{b}{3}\right)^2\)
Step 2: Simplify
\(\frac{a^2}{4} – \frac{b^2}{9}\)
Answer:\(\frac{a^2}{4} – \frac{b^2}{9}\)

Q4: Evaluate:

i. \((a+b)(a-b)(a^2+b^2)\)

Step 1: First, use the identity \((a+b)(a-b) = a^2 – b^2\)
\((a+b)(a-b)(a^2+b^2) = (a^2 – b^2)(a^2 + b^2)\)
Step 2: Apply the difference of squares again: \((p-q)(p+q) = p^2 – q^2\)
Here, \(p = a^2\), \(q = b^2\)
\((a^2 – b^2)(a^2 + b^2) = (a^2)^2 – (b^2)^2 = a^4 – b^4\)
Answer:\(a^4 – b^4\)

ii. \((3-2x)(3+2x)(9+4x^2)\)

Step 1: Use identity \((p+q)(p-q) = p^2 – q^2\) for the first two terms:
\((3-2x)(3+2x) = 3^2 – (2x)^2 = 9 – 4x^2\)
Step 2: Multiply by the third term:
\((9 – 4x^2)(9 + 4x^2)\)
Step 3: Apply difference of squares again:
\(9^2 – (4x^2)^2 = 81 – 16x^4\)
Answer:\(81 – 16x^4\)

iii. \((3x-4y)(3x+4y)(9x^2+16y^2)\)

Step 1: Use identity \((p+q)(p-q) = p^2 – q^2\) for the first two terms:
\((3x-4y)(3x+4y) = (3x)^2 – (4y)^2 = 9x^2 – 16y^2\)
Step 2: Multiply by the third term:
\((9x^2 – 16y^2)(9x^2 + 16y^2)\)
Step 3: Apply difference of squares again:
\((9x^2)^2 – (16y^2)^2 = 81x^4 – 256y^4\)
Answer:\(81x^4 – 256y^4\)

Q5: Use the formula: \(\left(a+b\right)\left(a-b\right)=a^2-b^2\) to evaluate:

i. 21 × 19

Step 1: Express numbers in the form \(a+b\) and \(a-b\):
21 = 20 + 1, 19 = 20 – 1
So, 21 × 19 = (20 + 1)(20 – 1)
Step 2: Apply the identity \((a+b)(a-b) = a^2 – b^2\)
(20 + 1)(20 – 1) = 20^2 – 1^2
Step 3: Simplify
400 – 1 = 399
Answer:399

ii. 33 × 27

Step 1: Express in the form \(a+b\) and \(a-b\):
33 = 30 + 3, 27 = 30 – 3
So, 33 × 27 = (30 + 3)(30 – 3)
Step 2: Apply identity
(30 + 3)(30 – 3) = 30^2 – 3^2
Step 3: Simplify
900 – 9 = 891
Answer:891

iii. 103 × 97

Step 1: Express in the form \(a+b\) and \(a-b\):
103 = 100 + 3, 97 = 100 – 3
So, 103 × 97 = (100 + 3)(100 – 3)
Step 2: Apply identity
(100 + 3)(100 – 3) = 100^2 – 3^2
Step 3: Simplify
10000 – 9 = 9991
Answer:9991

iv. 9.8 × 10.2

Step 1: Express in the form \(a+b\) and \(a-b\):
9.8 = 10 – 0.2, 10.2 = 10 + 0.2
So, 9.8 × 10.2 = (10 – 0.2)(10 + 0.2)
Step 2: Apply identity
(10 – 0.2)(10 + 0.2) = 10^2 – (0.2)^2
Step 3: Simplify
100 – 0.04 = 99.96
Answer:99.96

v. 7.7 × 8.3

Step 1: Express in the form \(a+b\) and \(a-b\):
7.7 = 8 – 0.3, 8.3 = 8 + 0.3
So, 7.7 × 8.3 = (8 – 0.3)(8 + 0.3)
Step 2: Apply identity
(8 – 0.3)(8 + 0.3) = 8^2 – (0.3)^2
Step 3: Simplify
64 – 0.09 = 63.91
Answer:63.91

Q6: Evaluate:

i. \( (6 – xy)(6 + xy) \)

Step 1: Apply identity \( (a+b)(a-b) = a^2 – b^2 \)
Step 2: \( (6 – xy)(6 + xy) = 6^2 – (xy)^2 \)
Answer:\( 36 – x^2y^2 \)

ii. \( \left(7x + \frac{2}{3}y\right)\left(7x – \frac{2}{3}y\right) \)

Step 1: Apply identity \( (a+b)(a-b) = a^2 – b^2 \)
Step 2: \( (7x + \frac{2}{3}y)(7x – \frac{2}{3}y) = (7x)^2 – (\frac{2}{3}y)^2 \)
Answer:\( 49x^2 – \frac{4}{9}y^2 \)

iii. \( \left(\frac{a}{2b} + \frac{2b}{a}\right)\left(\frac{a}{2b} – \frac{2b}{a}\right) \)

Step 1: Apply identity \( (a+b)(a-b) = a^2 – b^2 \)
Step 2: \( \left(\frac{a}{2b} + \frac{2b}{a}\right)\left(\frac{a}{2b} – \frac{2b}{a}\right) = \left(\frac{a}{2b}\right)^2 – \left(\frac{2b}{a}\right)^2 \)
Answer:\( \frac{a^2}{4b^2} – \frac{4b^2}{a^2} \)

iv. \( \left(3x – \frac{1}{2y}\right)\left(3x + \frac{1}{2y}\right) \)

Step 1: Apply identity \( (a+b)(a-b) = a^2 – b^2 \)
Step 2: \( (3x – \frac{1}{2y})(3x + \frac{1}{2y}) = (3x)^2 – (\frac{1}{2y})^2 \)
Answer:\( 9x^2 – \frac{1}{4y^2} \)

v. \( (2a+3)(2a-3)(4a^2+9) \)

Step 1: First, \( (2a+3)(2a-3) = (2a)^2 – 3^2 = 4a^2 – 9 \)
Step 2: Multiply by \( (4a^2 + 9) \): \( (4a^2 – 9)(4a^2 + 9) \)
Step 3: Apply difference of squares: \( (4a^2)^2 – 9^2 = 16a^4 – 81 \)
Answer:\( 16a^4 – 81 \)

vi. \( (a+bc)(a-bc)(a^2+b^2c^2) \)

Step 1: First, \( (a+bc)(a-bc) = a^2 – b^2c^2 \)
Step 2: Multiply by \( (a^2+b^2c^2) \): \( (a^2 – b^2c^2)(a^2 + b^2c^2) \)
Step 3: Apply difference of squares: \( (a^2)^2 – (b^2c^2)^2 = a^4 – b^4c^4 \)
Answer:\( a^4 – b^4c^4 \)

Q7: Expand:

i. \( \left(a + \frac{1}{2a}\right)^2 \)

Step 1: Apply identity \( (p+q)^2 = p^2 + 2pq + q^2 \), \( p = a, q = \frac{1}{2a} \)
Step 2: \( \left(a + \frac{1}{2a}\right)^2 = a^2 + 2\left(a \cdot \frac{1}{2a}\right) + \left(\frac{1}{2a}\right)^2 \)
Step 3: Simplify: \( a^2 + 1 + \frac{1}{4a^2}\)
Answer:\( a^2 + 1 + \frac{1}{4a^2} \)

ii. \( \left(2a – \frac{1}{a}\right)^2 \)

Step 1: Apply \( (p-q)^2 = p^2 – 2pq + q^2 \), \( p=2a, q = \frac{1}{a} \)
Step 2: \( (2a – \frac{1}{a})^2 = (2a)^2 – 2(2a \cdot \frac{1}{a}) + (\frac{1}{a})^2 \)
Step 3: Simplify: \( 4a^2 – 4 + \frac{1}{a^2} \)
Answer:\( 4a^2 – 4 + \frac{1}{a^2} \)

iii. \( (a+b-c)^2 \)

Step 1: Apply \( (p+q+r)^2 = p^2 + q^2 + r^2 + 2pq + 2pr + 2qr \), \( p=a, q=b, r=-c \)
Step 2: \( (a+b-c)^2 = a^2 + b^2 + (-c)^2 + 2ab + 2a(-c) + 2b(-c) \)
Step 3: Simplify: \( a^2 + b^2 + c^2 + 2ab – 2ac – 2bc \)
Answer:\( a^2 + b^2 + c^2 + 2ab – 2ac – 2bc \)

iv. \( (a-b+c)^2 \)

Step 1: Apply \( (p+q+r)^2 \), \( p=a, q=-b, r=c \)
Step 2: \( (a-b+c)^2 = a^2 + (-b)^2 + c^2 + 2a(-b) + 2a(c) + 2(-b)(c) \)
Step 3: Simplify: \( a^2 + b^2 + c^2 – 2ab + 2ac – 2bc \)
Answer:\( a^2 + b^2 + c^2 – 2ab + 2ac – 2bc \)

v. \( \left(3x + \frac{1}{3x}\right)^2 \)

Step 1: Apply \( (p+q)^2 = p^2 + 2pq + q^2 \), \( p=3x, q=\frac{1}{3x} \)
Step 2: \( (3x + \frac{1}{3x})^2 = (3x)^2 + 2(3x \cdot \frac{1}{3x}) + (\frac{1}{3x})^2 \)
Step 3: Simplify: \( 9x^2 + 2 + \frac{1}{9x^2} \)
Answer:\( 9x^2 + 2 + \frac{1}{9x^2} \)

Q8: Find the square of:

i. \( a + \frac{1}{5a} \)

Step 1: Apply identity \( (p+q)^2 = p^2 + 2pq + q^2 \), \( p = a, q = \frac{1}{5a} \)
Step 2: \( \left(a + \frac{1}{5a}\right)^2 = a^2 + 2\left(a \cdot \frac{1}{5a}\right) + \left(\frac{1}{5a}\right)^2 \)
Step 3: Simplify: \( a^2 + \frac{2}{5} + \frac{1}{25a^2} \)
Answer:\( a^2 + \frac{2}{5} + \frac{1}{25a^2} \)

ii. \( 2a – \frac{1}{a} \)

Step 1: Apply identity \( (p-q)^2 = p^2 – 2pq + q^2 \), \( p=2a, q=\frac{1}{a} \)
Step 2: \( \left(2a – \frac{1}{a}\right)^2 = (2a)^2 – 2(2a \cdot \frac{1}{a}) + \left(\frac{1}{a}\right)^2 \)
Step 3: Simplify: \( 4a^2 – 4 + \frac{1}{a^2} \)
Answer:\( 4a^2 – 4 + \frac{1}{a^2} \)

iii. \( x – 2y + 1 \)

Step 1: Apply identity \( (p+q+r)^2 = p^2 + q^2 + r^2 + 2pq + 2pr + 2qr \), \( p=x, q=-2y, r=1 \)
Step 2: \( (x – 2y + 1)^2 = x^2 + (-2y)^2 + 1^2 + 2(x \cdot -2y) + 2(x \cdot 1) + 2(-2y \cdot 1) \)
Step 3: Simplify: \( x^2 + 4y^2 + 1 – 4xy + 2x – 4y \)
Answer:\( x^2 + 4y^2 + 1 – 4xy + 2x – 4y \)

iv. \( 3a – 2b – 5c \)

Step 1: Apply \( (p+q+r)^2 \), \( p=3a, q=-2b, r=-5c \)
Step 2: \( (3a – 2b – 5c)^2 = (3a)^2 + (-2b)^2 + (-5c)^2 + 2(3a \cdot -2b) + 2(3a \cdot -5c) + 2(-2b \cdot -5c) \)
Step 3: Simplify: \( 9a^2 + 4b^2 + 25c^2 – 12ab – 30ac + 20bc \)
Answer:\( 9a^2 + 4b^2 + 25c^2 – 12ab – 30ac + 20bc \)

v. \( 2x + \frac{1}{x} + 1 \)

Step 1: Apply \( (p+q+r)^2 \), \( p=2x, q=\frac{1}{x}, r=1 \)
Step 2: \( (2x + \frac{1}{x} + 1)^2 = (2x)^2 + (\frac{1}{x})^2 + 1^2 + 2(2x \cdot \frac{1}{x}) + 2(2x \cdot 1) + 2(\frac{1}{x} \cdot 1) \)
Step 3: Simplify: \( 4x^2 + \frac{1}{x^2} + 1 + 4 + 4x + \frac{2}{x} = 4x^2 + \frac{1}{x^2} + 4x + \frac{2}{x} + 5 \)
Answer:\( 4x^2 + \frac{1}{x^2} + 4x + \frac{2}{x} + 5 \)

vi. \( 5 – x + \frac{2}{x} \)

Step 1: Apply \( (p+q+r)^2 \), \( p=5, q=-x, r=\frac{2}{x} \)
Step 2: \( (5 – x + \frac{2}{x})^2 = 5^2 + (-x)^2 + (\frac{2}{x})^2 + 2(5 \cdot -x) + 2(5 \cdot \frac{2}{x}) + 2(-x \cdot \frac{2}{x}) \)
Step 3: Simplify: \( 25 + x^2 + \frac{4}{x^2} – 10x + \frac{20}{x} – 4 = x^2 + \frac{4}{x^2} – 10x + \frac{20}{x} + 21 \)
Answer:\( x^2 + \frac{4}{x^2} – 10x + \frac{20}{x} + 21 \)

vii. \( 2x – 3y + z \)

Step 1: Apply \( (p+q+r)^2 \), \( p=2x, q=-3y, r=z \)
Step 2: \( (2x – 3y + z)^2 = (2x)^2 + (-3y)^2 + z^2 + 2(2x \cdot -3y) + 2(2x \cdot z) + 2(-3y \cdot z) \)
Step 3: Simplify: \( 4x^2 + 9y^2 + z^2 – 12xy + 4xz – 6yz \)
Answer:\( 4x^2 + 9y^2 + z^2 – 12xy + 4xz – 6yz \)

viii. \( x + \frac{1}{x} – 1 \)

Step 1: Apply \( (p+q+r)^2 \), \( p=x, q=\frac{1}{x}, r=-1 \)
Step 2: \( (x + \frac{1}{x} – 1)^2 = x^2 + (\frac{1}{x})^2 + (-1)^2 + 2(x \cdot \frac{1}{x}) + 2(x \cdot -1) + 2(\frac{1}{x} \cdot -1) \)
Step 3: Simplify: \( x^2 + \frac{1}{x^2} + 1 + 2 – 2x – \frac{2}{x} = x^2 + \frac{1}{x^2} – 2x – \frac{2}{x} + 3 \)
Answer:\( x^2 + \frac{1}{x^2} – 2x – \frac{2}{x} + 3 \)

Q9: Evaluate using expression of \(\left(a+b\right)^2\) or \(\left(a-b\right)^2\):

i. \( 208^2 \)

Step 1: Express \(208 = 200 + 8\)
Step 2: Apply identity \( (a+b)^2 = a^2 + 2ab + b^2 \), \( a=200, b=8 \)
Step 3: \( 208^2 = 200^2 + 2 \cdot 200 \cdot 8 + 8^2 \)
Step 4: \( 208^2 = 40000 + 3200 + 64 \)
Answer:43264

ii. \( 92^2 \)

Step 1: Express \(92 = 100 – 8\)
Step 2: Apply identity \( (a-b)^2 = a^2 – 2ab + b^2 \), \( a=100, b=8 \)
Step 3: \( 92^2 = 100^2 – 2 \cdot 100 \cdot 8 + 8^2 \)
Step 4: \( 92^2 = 10000 – 1600 + 64 \)
Answer:8464

iii. \( 9.4^2 \)

Step 1: Express \(9.4 = 10 – 0.6\)
Step 2: Apply identity \( (a-b)^2 = a^2 – 2ab + b^2 \), \( a=10, b=0.6 \)
Step 3: \( 9.4^2 = 10^2 – 2 \cdot 10 \cdot 0.6 + 0.6^2 \)
Step 4: \( 9.4^2 = 100 – 12 + 0.36 \)
Answer:88.36

iv. \( 20.7^2 \)

Step 1: Express \(20.7 = 20 + 0.7\)
Step 2: Apply identity \( (a+b)^2 = a^2 + 2ab + b^2 \), \( a=20, b=0.7 \)
Step 3: \( 20.7^2 = 20^2 + 2 \cdot 20 \cdot 0.7 + 0.7^2 \)
Step 4: \( 20.7^2 = 400 + 28 + 0.49 \)
Answer:428.49

Q10: Expand:

i. \( (2a+b)^3 \)

Step 1: Apply identity \( (p+q)^3 = p^3 + 3p^2q + 3pq^2 + q^3 \), \( p=2a, q=b \)
Step 2: \( (2a+b)^3 = (2a)^3 + 3(2a)^2b + 3(2a)b^2 + b^3 \)
Step 3: Simplify each term: \( 8a^3 + 12a^2b + 6ab^2 + b^3 \)
Answer:\(8a^3 + 12a^2b + 6ab^2 + b^3\)

ii. \( (a-2b)^3 \)

Step 1: Apply identity \( (p-q)^3 = p^3 – 3p^2q + 3pq^2 – q^3 \), \( p=a, q=2b \)
Step 2: \( (a-2b)^3 = a^3 – 3a^2(2b) + 3a(2b)^2 – (2b)^3 \)
Step 3: Simplify: \( a^3 – 6a^2b + 12ab^2 – 8b^3 \)
Answer:\(a^3 – 6a^2b + 12ab^2 – 8b^3\)

iii. \( (3x-2y)^3 \)

Step 1: Apply identity \( (p-q)^3 = p^3 – 3p^2q + 3pq^2 – q^3 \), \( p=3x, q=2y \)
Step 2: \( (3x-2y)^3 = (3x)^3 – 3(3x)^2(2y) + 3(3x)(2y)^2 – (2y)^3 \)
Step 3: Simplify: \( 27x^3 – 54x^2y + 36xy^2 – 8y^3 \)
Answer:\(27x^3 – 54x^2y + 36xy^2 – 8y^3\)

iv. \( (x+5y)^3 \)

Step 1: Apply identity \( (p+q)^3 = p^3 + 3p^2q + 3pq^2 + q^3 \), \( p=x, q=5y \)
Step 2: \( (x+5y)^3 = x^3 + 3x^2(5y) + 3x(5y)^2 + (5y)^3 \)
Step 3: Simplify: \( x^3 + 15x^2y + 75xy^2 + 125y^3 \)
Answer:\(x^3 + 15x^2y + 75xy^2 + 125y^3\)

v. \( \left(a+\frac{1}{a}\right)^3 \)

Step 1: Apply identity \( (p+q)^3 = p^3 + 3p^2q + 3pq^2 + q^3 \), \( p=a, q=\frac{1}{a} \)
Step 2: \( \left(a+\frac{1}{a}\right)^3 = a^3 + 3a^2(\frac{1}{a}) + 3a(\frac{1}{a})^2 + (\frac{1}{a})^3 \)
Step 3: Simplify: \( a^3 + 3a + \frac{3}{a} + \frac{1}{a^3} \)
Answer:\(a^3 + 3a + \frac{3}{a} + \frac{1}{a^3}\)

vi. \( \left(2a-\frac{1}{2a}\right)^3 \)

Step 1: Apply identity \( (p-q)^3 = p^3 – 3p^2q + 3pq^2 – q^3 \), \( p=2a, q=\frac{1}{2a} \)
Step 2: \( \left(2a-\frac{1}{2a}\right)^3 = (2a)^3 – 3(2a)^2(\frac{1}{2a}) + 3(2a)(\frac{1}{2a})^2 – (\frac{1}{2a})^3 \)
Step 3: Simplify: \( 8a^3 – 6a + \frac{3}{2a} – \frac{1}{8a^3} \)
Answer:\(8a^3 – 6a + \frac{3}{2a} – \frac{1}{8a^3}\)

Q11: Find the cube of:

i. \( (a+2)^3 \)

Step 1: Apply identity \( (p+q)^3 = p^3 + 3p^2q + 3pq^2 + q^3 \), \( p=a, q=2 \)
Step 2: \( (a+2)^3 = a^3 + 3a^2\cdot2 + 3a\cdot2^2 + 2^3 \)
Step 3: Simplify: \( a^3 + 6a^2 + 12a + 8 \)
Answer:\(a^3 + 6a^2 + 12a + 8\)

ii. \( (2a-1)^3 \)

Step 1: Apply identity \( (p-q)^3 = p^3 – 3p^2q + 3pq^2 – q^3 \), \( p=2a, q=1 \)
Step 2: \( (2a-1)^3 = (2a)^3 – 3(2a)^2\cdot1 + 3(2a)\cdot1^2 – 1^3 \)
Step 3: Simplify: \( 8a^3 – 12a^2 + 6a – 1 \)
Answer:\(8a^3 – 12a^2 + 6a – 1\)

iii. \( (2a+3b)^3 \)

Step 1: Apply identity \( (p+q)^3 = p^3 + 3p^2q + 3pq^2 + q^3 \), \( p=2a, q=3b \)
Step 2: \( (2a+3b)^3 = (2a)^3 + 3(2a)^2(3b) + 3(2a)(3b)^2 + (3b)^3 \)
Step 3: Simplify: \( 8a^3 + 36a^2b + 54ab^2 + 27b^3 \)
Answer:\(8a^3 + 36a^2b + 54ab^2 + 27b^3\)

iv. \( (3b-2a)^3 \)

Step 1: Apply identity \( (p-q)^3 = p^3 – 3p^2q + 3pq^2 – q^3 \), \( p=3b, q=2a \)
Step 2: \( (3b-2a)^3 = (3b)^3 – 3(3b)^2(2a) + 3(3b)(2a)^2 – (2a)^3 \)
Step 3: Simplify: \( 27b^3 – 54b^2a + 36ba^2 – 8a^3 \)
Answer:\(27b^3 – 54b^2a + 36ba^2 – 8a^3\)

v. \( \left(2x+\frac{1}{x}\right)^3 \)

Step 1: Apply identity \( (p+q)^3 = p^3 + 3p^2q + 3pq^2 + q^3 \), \( p=2x, q=\frac{1}{x} \)
Step 2: \( \left(2x+\frac{1}{x}\right)^3 = (2x)^3 + 3(2x)^2(\frac{1}{x}) + 3(2x)(\frac{1}{x})^2 + (\frac{1}{x})^3 \)
Step 3: Simplify: \( 8x^3 + 12x + \frac{6}{x} + \frac{1}{x^3} \)
Answer:\(8x^3 + 12x + \frac{6}{x} + \frac{1}{x^3}\)

vi. \( \left(x-\frac{1}{2}\right)^3 \)

Step 1: Apply identity \( (p-q)^3 = p^3 – 3p^2q + 3pq^2 – q^3 \), \( p=x, q=\frac{1}{2} \)
Step 2: \( \left(x-\frac{1}{2}\right)^3 = x^3 – 3x^2(\frac{1}{2}) + 3x(\frac{1}{2})^2 – (\frac{1}{2})^3 \)
Step 3: Simplify: \( x^3 – \frac{3}{2}x^2 + \frac{3}{4}x – \frac{1}{8} \)
Answer:\(x^3 – \frac{3}{2}x^2 + \frac{3}{4}x – \frac{1}{8}\)