CG Board Class 8 Mathematics Chapter 2 Solutions: Exponent (घातांक) – 2026-27

चूंकि घातांक 6 एक सम संख्या है, अतः (-1)⁶ = 1 होगा।

(-2)⁶ = [(-1) × 2]⁶ = (-1)⁶ × 2⁶ = 1 × 64 = 64

(d) (-3)⁶

चूंकि घातांक 6 एक सम संख्या है, अतः (-1)⁶ = 1 होगा।

(-3)⁶ = [(-1) × 3]⁶ = (-1)⁶ × 3⁶ = 1 × 729 = 729

(-5)² = [(-1) × 5]² = (-1)² × 5² = 1 × 5² = 5²

अब घातांक नियम a^m × aⁿ = a^m+n से:

5⁴ × 5² = 5⁴⁺² = 5⁶

(b) 15 × (-15)²⁵

(-15)²⁵ = [(-1) × 15]²⁵ = (-1)²⁵ × 15²⁵ = -1 × 15²⁵ = -(15)²⁵

अब, 15¹ × [-(15)²⁵] = -(15¹⁺²⁵) = -(15)²⁶

(c) 12⁵ ÷ (-12)³

(-12)³ = [(-1) × 12]³ = (-1)³ × 12³ = -1 × 12³ = -(12)³

अब भाग नियम के प्रयोग से:

12⁵ ÷ [-(12)³] = -(12^5-3) = -(12)²

(d) (-p)¹⁴ ÷ (-p)⁷

भाग नियम a^m ÷ aⁿ = a^m-n से:

(-p)¹⁴ ÷ (-p)⁷ = (-p)^14-7 = (-p)⁷ = -(p)⁷

वाम पक्ष (LHS):** (-2)⁴ × (-2)² = (-2)⁴⁺² = (-2)⁶ = 64

• दक्षिण पक्ष (RHS): (-2)⁸ ÷ (-2)² = (-2)^8-2 = (-2)⁶ = 64

चूंकि LHS = RHS = 64, अतः कथन सत्यापित हुआ।

(b) (-3)² × (-3)^-6 = (1)/([(-3)²]²)

• वाम पक्ष (LHS): (-3)² × (-3)^-6 = (-3)^{2 + (-6)} = (-3)^-4 = (1)/((-3)⁴) = 1/81
• दक्षिण पक्ष (RHS): (1)/([(-3)²]²) = (1)/((-3)^{2 × 2}) = (1)/((-3)⁴) = 1/81

चूंकि LHS = RHS = 1/81, अतः कथन सत्यापित हुआ।

(c) (-7)³² ÷ (-7)³² = 1

• वाम पक्ष (LHS): (-7)³² ÷ (-7)³² = (-7)^32-32 = (-7)⁰ = 1
• दक्षिण पक्ष (RHS): 1

चूंकि LHS = RHS = 1, अतः कथन सत्यापित हुआ।

घातांक योग नियम (a/b)^m × (a/b)ⁿ = (a/b)^m+n से:

(1/7)²⁺¹ = (1/7)³ = 1/343

(b) (4/5)⁴ × (5/4)²

ऋणात्मक घातांक नियम से आधार 5/4 को 4/5 के रूप में बदलने पर:

(5/4)² = (4/5)^-2

अब, (4/5)⁴ × (4/5)^-2 = (4/5)^4-2 = (4/5)² = 16/25

(c) (-5)³ ÷ (-5)^-2

भाग नियम a^m ÷ aⁿ = a^m-n से:

(-5)^{3 – (-2)} = (-5)^{3 + 2} = (-5)⁵ = -3125

(d) (3/4)³ × (3/4)^-5

घातांक योग नियम से:

(3/4)^{3 + (-5)} = (3/4)^-2 = (4/3)² = 16/9

-25/49 = -5²/7² = -(5/7)²

(b) 27/125

27/125 = 3³/5³ = (3/5)³

(c) 64/729

64/729 = 2⁶/3⁶ = (2/3)⁶

वाम पक्ष (LHS):**

पहला पद: (5/7 × 7/5)⁷ = 1⁷ = 1

दूसरा पद: (19/3 × 3/19)² = 1² = 1

LHS = 1 – 1 = 0

• दक्षिण पक्ष (RHS): 0

चूंकि LHS = RHS = 0, अतः सिद्ध हुआ।

(b) (p/q)^m × (p/q)^m × (q/p)^m = (q/p)^-m

• वाम पक्ष (LHS):

(p/q)^m × (p/q)^m × (p/q)^-m = (p/q)^{m + m – m} = (p/q)^m

व्युत्क्रम घातांक नियम (p/q)^m = (q/p)^-m से:

LHS = (q/p)^-m

• दक्षिण पक्ष (RHS): (q/p)^-m

चूंकि LHS = RHS, अतः सिद्ध हुआ।

(c) (25/16)^-4 = (16/25)⁴

• वाम पक्ष (LHS):

ऋणात्मक घातांक नियम (a/b)^-m = (b/a)^m के प्रयोग से:

(25/16)^-4 = (16/25)⁴

• दक्षिण पक्ष (RHS): (16/25)⁴

चूंकि LHS = RHS, अतः सिद्ध हुआ।

(a) (5/4)^-65 = (4/5)⁶⁵

• Statement Analysis: सत्य [Rule: (a/b)^-m = (b/a)^m]

(b) (32/19)^-150 = (19/32)¹⁵⁰

• Statement Analysis: सत्य [Rule: (a/b)^-m = (b/a)^m]

(c) (25 × 3)⁵ = 25 × 3⁵

• Statement Analysis: असत्य [Correct expansion is (25 × 3)⁵ = 25⁵ × 3⁵]

(d) (27/16)^-15 = (27/16)¹⁵

• Statement Analysis: असत्य [Correct form is (27/16)^-15 = (16/27)¹⁵]