Rameshwari School Class 7 Maths Solution — অনুশীলনী: 1E (পৰিমেয় সংখ্যা আৰু সূচক)

 ৰামেশ্বৰী বিদ্যালয়ৰ সপ্তম শ্ৰেণীৰ গণিত পাঠ্যপুথিৰ অনুশীলনী: 1E

Class 7 Maths Solution — অনুশীলনী: 1E (পৰিমেয় সংখ্যা আৰু সূচক)

প্ৰশ্ন 1: ঘাত সূত্ৰ প্ৰয়োগ কৰি মান উলিওৱাঁঃ

[i] মান উলিওৱা:
• 8³ = 8 × 8 × 8 = 512
• 15² = 15 × 15 = 225
• (-2)³ = (-2) × (-2) × (-2) = -8
• a² · a⁵ = a²⁺⁵ = a⁷
• a²(ab⁵)b³ = a²⁺¹ · b⁵⁺³ = a³b⁸

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[ii] x⁵ × (xy³) × (x³y⁴)
= x⁵⁺¹⁺³ · y³⁺⁴
= x⁹y⁷

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[iii] 9⁰ × (12)⁰
= 1 × 1 = 1

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[iv] (2⁴)³ × 2⁶ × 2³
= 2¹² × 2⁶ × 2³
= 2¹²⁺⁶⁺³
= 2²¹

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[v] (a²b³)³ × (ab³)⁴
= a⁶b⁹ × a⁴b¹²
= a⁶⁺⁴ · b⁹⁺¹²
= a¹⁰b²¹

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[vi] (xy²)³ × (x²y³)⁴ × (x³y²)³
= x³y⁶ × x⁸y¹² × x⁹y⁶
= x³⁺⁸⁺⁹ · y⁶⁺¹²⁺⁶
= x²⁰y²⁴

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[vii] 9³ × 3⁴ × 3⁶
= (3²)³ × 3⁴⁺⁶
= 3⁶ × 3¹⁰
= 3¹⁶

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[viii] 1⁰ × 5¹⁰ × 25⁵ × 125²
= 1 × 5¹⁰ × (5²)⁵ × (5³)²
= 5¹⁰ × 5¹⁰ × 5⁶
= 5^10+10+6
= 5²⁶

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[ix] 7⁴ × 7³ × 7⁷
= 7⁴⁺³⁺⁷
= 7¹⁴

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[x] 4³ × 2² × 8⁶
= (2²)³ × 2² × (2³)⁶
= 2⁶ × 2² × 2¹⁸
= 2⁶⁺²⁺¹⁸
= 2²⁶

প্ৰশ্ন 2: পৰিমেয় সংখ্যাৰ ঘাত হিচাপে প্ৰকাশ কৰাঁঃ

[i] 4⁷ ÷ 4³
= 4^7-3 = 4⁴

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[ii] (-12)⁴ (-12)³ (-12)¹⁰
= (-12)⁴⁺³⁺¹⁰ = (-12)¹⁷

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[iii] 7³ × 49
= 7³ × 7² = 7⁵

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[iv] 2¹ × 3³ × 5³10 × 9
= 2 × 3³ × 5³(2 × 5) × 3²
= 2^1-1 × 3^3-2 × 5^3-1
= 1 × 3¹ × 5² = 3 × 5²

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[v] (-13)³ × (-13)⁴ × (-13)³
= (-13)³⁺⁴⁺³ = (-13)¹⁰

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[vi] (12)³ × (12)⁵ × (12)^-1
= 12^3+5-1 = 12⁷

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[vii] ((-13)⁶)⁵
= (-13)^{6 × 5} = (-13)³⁰

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[viii] (-43) × (-43)⁶ × (-43)^-2
= (-43)^1+6-2 = (-43)⁵

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[ix] (16)⁴ (16)^-3 ((16)³)⁷
= (16)^{4 - 3 + 21} = (16)²²

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[x] (14) (14)^-9 × (14)⁷
= (14)^{1 - 9 + 7} = (14)^-1

প্ৰশ্ন 3: বিয়োগ সূচক আঁতৰাই প্ৰকাশ কৰাঁঃ

[i] 4^-15^-1
= 5¹4¹ = 54

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[ii] (16)^-1 (3^-1 × 5^-1)
= 6 × (13 × 15)
= 6 × 115 = 615 = 25

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[iii] অতিৰিক্ত মানসমূহ:
• (-17)^-2 = (-7)² = 49

• (78)^-3 = (87)³

• x^-5(y^-5)² = x^-5y^-10 = y¹⁰x⁵

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[iv] প্ৰকাশ কৰা:
• (a³b^-2)^-2 = a^-6b⁴ = b⁴a⁶

• (a⁶b⁴c^-3)^-5 = (a⁶ b⁴c³)^-5 = a^-30b^-20c^-15 = 1a³⁰b²⁰c¹⁵

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[v] প্ৰকাশ কৰা:
• [(79)^-1]³ = (79)^-3 = (97)³

• (811)^-4 = (118)⁴

প্ৰশ্ন 4: উচ্চতম ঘাতত প্রকাশ কৰাঃ

[i] 81 = 3⁴

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[ii] 49 = 7²

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[iii] 125216 = 5³6³ = (56)³

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[iv] 2516 = 5²4² = (54)²

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[v] 3431331 = 7³11³ = (711)³

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[vi] -49 × 49125 × 5 = -7² × 7²5³ × 5¹ = -7⁴5⁴ = -(75)⁴

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[vii] 7291000 = 9³10³ = (910)³

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[viii] 289361 = 17²19² = (1719)²

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[ix] 841 = 29²

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[x] 64729 = 2⁶3⁶ = (23)⁶

প্ৰশ্ন 5: সৰল কৰাঃ

[i] 13(27 - 13)³ × 7³
= 13(-121)³ × 7³
= 13 × [-13³ × 7³] × 7³
= -13 × 27 = -181

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[ii] (5²6 × 19)(12 × 52)²
= [2554] × [54]²
= 2554 × 2516 = 625864

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[iii] x⁰ + (x+1)⁰ - (x-2)
= 1 + 1 - (x - 2)
= 2 - x + 2 = 4 - x

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[iv] [(25)⁸ × (15)^-6] ÷ [(25)⁴ × (15)^-2]
= [(25)⁸ × 5⁶] ÷ [(25)⁴ × 5²]
= (25)^8-4 × 5^6-2
= (25)⁴ × 5⁴ = (25 × 5)⁴ = 2⁴ = 16

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[v] [5 + 13(38)⁰]^-4
= [5 + 13 × 1]^-4
= (163)^-4 = (316)⁴ = 8165536

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[vi] (a³b²)³ × (a⁴b²)⁵ × (c²a²)²a² b²c³
= a⁹b⁶ × a²⁰b¹⁰ × c⁴a⁴a²b² c³
= a³³b¹⁶c⁴a²b²c³
= a^33-2b^16-2c^4-3 = a³¹b¹⁴c

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[vii] (4a³b²)³ × (5a² bc)²2 × 20 × (ab)⁴
= 64a⁹b⁶ × 25a⁴b²c²40a⁴b⁴
= 1600a¹³b⁸c²40a⁴b⁴
= 40a^13-4b^8-4c² = 40a⁹b⁴c²

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[viii] (19 ÷ 116) × (13)⁷ × (15)² × (12)³
= (169) × [13⁷ × 5² × 2³]
= 2⁴3² × 3⁷ × 5² × 2³
= 2^4-3 ÷ (3²⁺⁷ × 5²) = 23⁹ × 5²

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[ix] (43 × 1627)² × [(34) ÷ (23)⁴]
= (6481)² × [34 × 8116]
= (2⁶3⁴)² × [3⁵2⁶]
= (2¹²3⁸) × (3⁵2⁶) = 2^12-63^8-5 = 2⁶3³ = 6427

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[x] [(110)^-2 ÷ (10)^-4] × [(110)^-5]^-2
= [10² ÷ 10^-4] × (110)¹⁰
= 10²⁺⁴ × 10^-10
= 10⁶ × 10^-10 = 10^-4 = 110⁴ (বা 110000)

প্ৰশ্ন 6: গুণাত্মক বিপৰীতবোৰ লিখাঁঃ

• 5⁴-ৰ গুণাত্মক বিপৰীত = 15⁴ = 5^-4
• (-13)³-ৰ গুণাত্মক বিপৰীত = (-3)³ = -27
• 1a⁵-ৰ গুণাত্মক বিপৰীত = a⁵
• (-xy)⁵-ৰ গুণাত্মক বিপৰীত = (-yx)⁵
• 6^-5-ৰ গুণাত্মক বিপৰীত = 6⁵

প্ৰশ্ন 7: (625576)^-3-ৰ লগত কিমান পূৰণ কৰিলে 4850 হ'b?

ধৰা হ'ল নিৰ্ণেয় সংখ্যাটো k।
k × (576625)³ = 2425
⇒ k × (24²25²)³ = 2425
⇒ k × (2425)⁶ = 2425
⇒ k = (2425)^1-6 = (2425)^-5 = (2524)⁵

প্ৰশ্ন 8: কোন সংখ্যাক (34)³-ৰে ভাগ কৰিলে (-23)⁶ পাবা?

ধৰা হ'ল সংখ্যাটো x।
x ÷ (34)³ = (-23)⁶
⇒ x = (2/3)⁶ × (34)³
⇒ x = 2⁶3⁶ × 3³2⁶ = 13³ = 127

প্ৰশ্ন 9: 8y³3x⁴-ৰ তৃতীয় ঘাতৰ লগত (3x³-2y)²-ৰ চতুৰ্থ ঘাতক পূৰণ কৰাঁ।

= (8y³3x⁴)³ × [(3x³-2y)²]⁴
= (2⁹y⁹3³x¹²) × (3⁸ x²⁴2⁸y⁸)
= 2^9-8 × 3^8-3 × x^24-12 × y^9-8
= 2 × 243 × x¹²y = 486x¹²y

প্ৰশ্ন 10: যদি m² × m^-4 = m^2x, তেন্তে x-ৰ মান কিমান?

m^2-4 = m^2x
⇒ m^-2 = m^2x
⇒ -2 = 2x ⇒ x = -1

প্ৰশ্ন 11: যদি (23)^-x = (23)^x তেন্তে x-ৰ মান কিমান?

-x = x
⇒ 2x = 0 ⇒ x = 0

প্ৰশ্ন 12: 2^m = 1024 হ'লে m-ৰ মান উলিওৱা।

2^m = 2¹⁰
⇒ m = 10

প্ৰশ্ন 13: 52-ৰ লগত কি পূৰণ কৰিলে 62510000 হ'b?

ধৰা হ’ল সংখ্যাটো = x
x × (52) = 116
⇒ x = 116 × 25 = 140

প্ৰশ্ন 14: a³ b²ca-ক কিহেৰে ভাগ কৰিলে cb পাবা?

ধৰা হ’ল সংখ্যাটো = x
(a²b²c) ÷ x = cb
⇒ x = (a²b²c) × (bc) = a²b³c²

প্ৰশ্ন 15: 4^x · 2^x+2 = 32 হ'লে x-ৰ মান নিৰ্ণয় কৰা।

2^2x · 2^x+2 = 2⁵
⇒ 2^3x+2 = 2⁵
⇒ 3x + 2 = 5
⇒ 3x = 3 ⇒ x = 1

প্ৰশ্ন 16: যদি (29)⁶ ÷ (29)³ = (29)^m-1m+1 হয়, তেন্তে m = ?

6 - 3 = m - 1m + 1
⇒ 3 = m - 1m + 1
⇒ 3m + 3 = m - 1
⇒ 2m = -4 ⇒ m = -2

প্ৰশ্ন 17: যদি (67)^5+2x × (67)^11-x = (67)^8x-5 হয়, তেন্তে x = ?

(5 + 2x) + (11 - x) = 8x - 5
⇒ 16 + x = 8x - 5
⇒ 7x = 21 ⇒ x = 3

প্ৰশ্ন 18: যদি 5^2p+1 ÷ 25 = 125 হয়, তেন্তে p = ?

5^2p+1 ÷ 5² = 5³
⇒ 5^2p-1 = 5³
⇒ 2p - 1 = 3
⇒ 2p = 4 ⇒ p = 2

প্ৰশ্ন 19: যদি a=2, আৰু b=3 হয় তেন্তে তলৰ ৰাশি সমূহৰ মান উলিওৱা।

[i] (a+b)^a = (2+3)² = 5² = 25

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[ii] (ab)^b = (2 × 3)³ = 6³ = 216

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[iii] (ba)^a = (32)² = 94

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[iv] (a-b)^b = (2-3)³ = (-1)³ = -1

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[v] (ab)^b = (23)³ = 827

প্ৰশ্ন 20: সৰল ৰূপত প্ৰকাশ কৰা:

(x^ax^b)⁴ × (x^bx^c)³ × (x^cx^a)²
= x^4(a-b) × x^3(b-c) × x^2(c-a)
= x^{4a - 4b + 3b - 3c + 2c - 2a}
= x^{2a - b - c}

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