Class 8 Maths · Chapter 1
Short answer: Ganita Prakash Class 8 Maths Part 1, Chapter 1 "A Square and A Cube" squares, cubes, square roots, cube roots aur unki properties cover karta hai — patterns, perfect squares/cubes identify karna, aur prime factorisation method se square root/cube root nikalna sikhata hai. Ye foundation chapter hai jo aage ke number-theory aur algebra chapters (jaise "We Distribute Yet Things Multiply" aur "Fractions in Disguise") ke liye base banata hai.
NCERT ne 2026-27 session se Class 8 Maths ko naye Ganita Prakash format me rationalise kar diya hai — ab ye ek single thick book nahi, balki Part 1 aur Part 2 me split hai, total 14 chapters ke saath (Part 1: chapters 1-7, Part 2: chapters 8-14). Ye naya NCF-aligned curriculum hai jo purani book se kaafi different feel deta hai — zyada activity-based, kam rote-formula.
Chapter 1, "A Square and A Cube," Part 1 ka opening chapter hai aur poori number-sense building ki neev hai. Is chapter me students squares aur cubes ke patterns explore karte hain — kaise consecutive odd numbers ka sum perfect square banata hai, square numbers ki digit patterns, aur prime factorisation method se square root aur cube root nikalna. Ye sirf ek standalone topic nahi hai — is chapter ki understanding directly Part 1 ke aage ke chapters jaise "Proportional Reasoning 1 vs 2," "We Distribute Yet Things Multiply," aur "Fractions in Disguise" me kaam aati hai, jahan number properties baar baar use hoti hain.
Ganita Prakash class 8 maths chapter 1 solutions is page par CBSE marking-scheme style step-by-step working ke saath diye gaye hain, taaki students exam me full marks score kar sakein.
Chapter 1 Summary — 5 Minute Revision
Chapter 1 "A Square and A Cube" ke core ideas:
- Perfect square — koi bhi number jo kisi natural number ka square ho (jaise 4, 9, 16, 25)
- Perfect cube — koi bhi number jo kisi natural number ka cube ho (jaise 8, 27, 64, 125)
- Patterns in squares — consecutive odd numbers ka sum, square numbers ke unit digit patterns, aur square between consecutive squares me non-perfect-square numbers ki ginti
- Prime factorisation method — square root aur cube root nikalne ka systematic tareeka, jisme prime factors ko pairs (square ke liye) ya triplets (cube ke liye) me group kiya jata hai
- Properties — perfect squares kabhi 2, 3, 7, ya 8 pe end nahi hote; perfect square ka unit digit hamesha 0,1,4,5,6,9 hota hai
Ye concepts sirf is chapter tak limited nahi — Ganita Prakash Part 1 ke "Number Play" (chapter 5) aur "Algebra Play" jaise chapters me bhi number properties ka yahi foundational logic reuse hota hai.
In-Text Questions — Solutions
Kya 225 ek perfect square hai? Prime factorisation method se check karo.
225 ka prime factorisation karte hain:
225 = 3 × 3 × 5 × 5 = 32 × 52
Yahan har prime factor pair me hai (3 do baar, 5 do baar) — koi factor akela nahi bacha. Isliye 225 ek perfect square hai.
√225 = 3 × 5 = 15
1 se 100 ke beech kitne perfect square numbers hain?
1² se lekar 10² tak ke squares 1 se 100 ke range me aate hain, kyunki 10² = 100 aur 11² = 121 (jo range se bahar hai).
Isliye 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 — total 10 perfect squares hain.
64 ka cube root prime factorisation method se nikalo.
64 = 2 × 2 × 2 × 2 × 2 × 2 = 23 × 23
Prime factors ko triplets me group karte hain: (2×2×2) × (2×2×2)
∛64 = 2 × 2 = 4
Exercise Questions — Solutions (Q1–Q6)
Q1. Bina calculation kiye batao ki kya 2352 ek perfect square hai. Reason do.
Step 1: 2352 ka prime factorisation karte hain.
2352 = 2 × 2 × 2 × 2 × 3 × 7 × 7 = 24 × 31 × 72
Step 2: Pairs check karte hain — 2 char baar (2 pairs), 7 do baar (1 pair), lekin 3 sirf ek baar hai — akela bach jata hai.
Answer: Kyunki 3 ka koi pair nahi hai, 2352 ek perfect square NAHI hai. (1 mark reasoning + 1 mark conclusion)
Q2. Sabse chota number kaunsa hai jisse 2352 ko multiply karne par woh perfect square ban jaye?
Q1 se hume pata hai: 2352 = 24 × 31 × 72
Sirf 3 hi unpaired hai, isliye 2352 ko 3 se multiply karna hoga taaki 3 bhi pair ban jaye.
2352 × 3 = 7056 = 24 × 32 × 72 = (22 × 3 × 7)2 = 842
Answer: 3
Q3. Consecutive odd numbers ka pattern use karke 62 ko sum of odd numbers ke roop me likho.
Rule: n2 = pehle n odd numbers ka sum.
n = 6 ke liye, pehle 6 odd numbers chahiye: 1, 3, 5, 7, 9, 11
1 + 3 + 5 + 7 + 9 + 11 = 36 = 62
Verified: 6² = 36 ✓
Q4. 1728 ka cube root prime factorisation se nikalo, poora working dikhao.
Step 1: Prime factorisation.
1728 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3 = 26 × 33
Step 2: Triplets me group karo: (2×2×2) × (2×2×2) × (3×3×3)
Step 3: Har triplet se ek factor lo.
∛1728 = 2 × 2 × 3 = 12
Answer: 12 (2 marks working + 1 mark final answer)
Q5. Baudhayana ke Pythagoras-theorem jaisa result kya bataata hai, aur square numbers se uska kya connection hai?
Baudhayana Shulba Sutra (Pythagoras se pehle ka Indian text) me ek geometric result diya gaya hai jo aaj Pythagoras theorem ke naam se jaana jaata hai — right-angled triangle me hypotenuse ka square, baaki do sides ke squares ke sum ke barabar hota hai.
(hypotenuse)2 = (side 1)2 + (side 2)2
Example: 3-4-5 triangle me, 32 + 42 = 9 + 16 = 25 = 52. Isliye square numbers ka geometric interpretation (area of a square) is theorem ki foundation hai — yehi wajah hai chapter square numbers se shuru hota hai, taaki students area-based reasoning samjhein jo aage geometry chapters me kaam aayega.
Q6. 1000 ka cube root kya hai — verify karo ki 1000 perfect cube hai.
Step 1: Prime factorisation.
1000 = 2 × 2 × 2 × 5 × 5 × 5 = 23 × 53
Step 2: Dono factors clean triplets me hain, isliye 1000 ek perfect cube hai.
∛1000 = 2 × 5 = 10
Verification: 10 × 10 × 10 = 1000 ✓
Important Equations — Ek Nazar Me
| Rule / Formula | Explanation |
|---|---|
| n2 = sum of first n odd numbers | Perfect squares consecutive odd numbers ke sum se banate hain |
| Between n2 and (n+1)2 → 2n non-square numbers | Do consecutive perfect squares ke beech non-perfect-square numbers ki count |
| Unit digit of perfect square ∈ {0,1,4,5,6,9} | 2, 3, 7, 8 pe kabhi perfect square end nahi hota |
| √(a × b) = √a × √b | Prime factorisation method — pairs banao, ek factor har pair se lo |
| ∛(a × b) = ∛a × ∛b | Cube root ke liye triplets banao, ek factor har triplet se lo |
| a2 = b2 + c2 (Baudhayana/Pythagoras result) | Right-angled triangle me square area ka relation |
↔ Table ko side me swipe karein
Common Mistakes — Yahan Marks Kat te Hain
- Prime factorisation me factor tree galat banana — kisi prime factor ko miss kar dena, jisse pairs/triplets ka count wrong ho jata hai
- Square root nikalte waqt sirf ek factor le lena jab prime factor higher power (jaise 24) me ho — 24 ka matlab do pairs hain, na ki ek
- Perfect square aur perfect cube ke unit-digit rules ko mix kar dena — dono ke rules alag hain
- Sum of consecutive odd numbers wala pattern likhte waqt n odd numbers ki jagah kam ya zyada numbers le lena
- 'Smallest number to multiply/divide for perfect square' wale questions me sabhi unpaired factors identify na karna — sirf ek factor dekh kar answer likh dena
- Cube root nikalte waqt triplets ki jagah galti se pairs bana dena (square root ka method cube root pe apply kar dena)
Board-Style Important Questions
- Board/school exam style: 'Bina calculation kiye batao ki 2 marks ke liye kya diya gaya number perfect square/cube hai — reason do' — ye question format Ganita Prakash ke pattern se directly aata hai
- 'Sabse chota number nikalo jisse multiply/divide karne par given number perfect square/cube ban jaye' — prime factorisation based application question, common in school tests
- Consecutive odd numbers ka sum use karke kisi given n2 ko express karna — pattern-based conceptual question
- Do consecutive perfect squares ke beech non-square numbers ki ginti nikalna (formula 2n use karke)
- Word problem: area-based context (jaise square plot/field) me square root nikalna real-life application ke roop me
- Baudhayana ke Pythagoras-jaisa result explain karne wala short-answer conceptual question, history-of-math angle se
Aksar Poochhe Jaane Wale Sawaal
Class 8 maths ki new book ka naam kya hai 2026-27 session ke liye?
NCERT ne Class 8 Maths ki new book ka naam Ganita Prakash rakha hai, jo do parts me aati hai — Part 1 (chapters 1-7) aur Part 2 (chapters 8-14). Ye purani single-book format se rationalise ho kar aayi hai 2026-27 session ke liye.
Ganita Prakash Part 1 aur Part 2 me kitne chapters hain, aur list kya hai?
Total 14 chapters hain. Part 1 me chapters 1 se 7 tak hain, jinme se Chapter 1 hai 'A Square and A Cube' aur Chapter 5 hai 'Number Play'. Part 2 me chapters 8 se 14 tak hain. Exact chapter-by-chapter naam order NCERT ki official website (ncert.nic.in) par verify kiya ja sakta hai.
Class 8 maths ki old book aur Ganita Prakash me kya difference hai?
Sabse bada structural difference ye hai ki purani book single volume thi, jabki Ganita Prakash do parts me split hai. Content approach bhi zyada activity-based aur pattern-discovery-driven hai, rote-formula-heavy approach ke bajaye — jaise Chapter 1 me squares/cubes seekhne ka tareeka patterns explore karke hai, seedha formula ratt kar nahi.
Baudhayana ka Pythagoras theorem se kya connection hai jo Chapter 1 me discuss hota hai?
Baudhayana ek ancient Indian mathematician the jinke Shulba Sutra me ek geometric result mila hai jo aaj Pythagoras theorem ke naam se famous hai — right-angled triangle ke hypotenuse ka square, baaki do sides ke squares ke sum ke barabar hota hai. Chapter 1 me square numbers ka geometric (area-based) interpretation isi context ko samajhne ki neev banata hai.
Prime factorisation method se square root aur cube root nikalne me kya farak hai?
Square root ke liye prime factors ko pairs (do-do ke groups) me group karte hain, aur cube root ke liye triplets (teen-teen ke groups) me. Har pair/triplet se ek factor le kar unhe multiply karne se root mil jata hai.
Kya 'Proportional Reasoning' wala chapter aur Chapter 1 connected hain?
Directly nahi, lekin dono hi Ganita Prakash Part 1 ke number-sense-building chapters hain. Chapter 1 me squares/cubes ki understanding number properties ke prime-factorisation-based reasoning develop karti hai, jo Proportional Reasoning aur baad ke algebra-based chapters (jaise 'We Distribute Yet Things Multiply' aur 'Algebra Play') me bhi kaam aati hai jahan number patterns aur relationships analyse karne hote hain.
Class 8 Maths — Saare Chapters
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