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ICSE Class 8 Mid-term 2026 : Mathematics

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NCERT Solutions For Class 8 Maths Chapter 2 - Power Play 1) Express the following in exponential form (i) 6 6 6 6 Answer: 64 (ii) y y Answer: y2 (iii) b b b b Answer: b4 (iv) 5 5 7 7 7 Answer: 52 73 (v) 2 2 a a Answer: 22 a2 (vi) a a a c c c c d Answer: a3 c4 d Extra: Values with Base 0 What is 02 and 05? Answer: 02 = 0 and 05 = 0. www.vedantu.com 1 What is 0n (for positive integer n)? Answer: 0n = 0 for all positive integers n. (Note: 00 is undefined/not defined.) How do you read 54? Answer: 5 raised to the power 4 / 5 to the power 4 / 5 power 4 / 4th power of 5 . 2) Express each number as a product of powers of prime factors (exponential form) (i) 648 Answer: 648 = 23 34 (ii) 405 Answer: 405 = 34 5 (iii) 540 Answer: 540 = 22 33 5 (iv) 3600 Answer: 3600 = 24 32 52 3) Write the numerical value Note: Interpreting the given expressions in standard exponent form. (i) 2 103 Answer: 2 1000 = 2000 www.vedantu.com 2 (ii) 72 23 Answer: 49 8 = 392 (iii) 3 44 Answer: 3 256 = 768 (iv) ( 3)2 ( 5)2 Answer: 9 25 = 225 (v) 32 104 Answer: 9 10000 = 90000 (vi) ( 2)5 ( 10)6 Answer: ( 32) 1,000,000 = 32,000,000 Class 8 Powers & Exponents 1. Find out the units digit in the value of 2224 432? [Hint: 4 = 22] Answer: 432 = (22)32 = 264. Hence 2224 432 = 2224 64 = 2160. Units digit of powers of 2 cycles 2,4,8,6 every 4. Since 160 0 (mod 4), the units digit is 6. 2. There are 5 bottles in a container. Every day, a new container is brought in. How many bottles would be there after 40 days? Answer: Each day adds 5 bottles. In 40 days: 40 5 = 200 bottles. 3. Write the given number as the product of two or more powers in three different ways. The powers can be any integers. (i) 643 (ii) 1928 (iii) 32 5 www.vedantu.com 3 Answer: (i) 643 (26)3 = 218 = 210 28 49 = 44 44 4 86 = 83 83 (ii) 1928 1914 1914 1920 198 1910 1910 198 (iii) 32 5 (25) 5 = 2 25 = 2 10 2 10 2 5 8 5 4 5 16 5 2 5 4. Examine each statement below and find out if it is Always True , Only Sometimes True , or Never True . Explain your reasoning. (i) Cube numbers are also square numbers. (ii) Fourth powers are also square numbers. (iii) The fifth power of a number is divisible by the cube of that number. (iv) The product of two cube numbers is a cube number. (v) q46 is both a 4th power and a 6th power (q is a prime number). Answer: (i) Only Sometimes True (e.g., 64 is both; but 8 is not a square). (ii) Always True since n4 = (n2)2. www.vedantu.com 4 (iii) Always True (n5 n3 = n2). (iv) Always True (a3 b3 = (ab)3). (v) Never True because 46 is not a multiple of 4 or 6 (LCM = 12). 5. Simplify and write these in the exponential form. (i) 10 2 10 5 (ii) 57 54 (iii) 9 7 94 (iv) (13 2) 3 (v) m5/n12 (mn)9 Answer: (i) 10 7 (ii) 53 (iii) 9 11 (iv) 136 (v) m14 n 3 6. If 122 = 144 what is (i) (1.2)2 (ii) (0.12)2 (iii) (0.012)2 (iv) 1202 Answer: (i) 1.44 (ii) 0.0144 (iii) 0.000144 (iv) 14,400 www.vedantu.com 5 7. Circle the numbers that are the same 24 36 64 32 610 182 62 624 Answer: The equal ones are 24 36, 64 32, and 182 62 (all simplify to 2436). The others differ. 8. Identify the greater number in each of the following (i) 43 or 34 (ii) 28 or 82 (iii) 1002 or 2100 Answer: (i) 34 (81 > 64) (ii) 28 (256 > 64) (iii) 2100 (much larger than 10,000) 9. A dairy plans to produce 8.5 billion packets of milk in a year. They want a unique ID (identifier) code for each packet. If they choose to use the digits 0 9, how many digits should the code consist of? Answer: With digits only, codes of length d give 10d possibilities. Need 10d 8.5 109 smallest d is 10. 10. 64 is a square number (82) and a cube number (43). Are there other numbers that are both squares and cubes? Is there a way to describe such numbers in general? Answer: Yes. Numbers that are both squares and cubes are exactly the sixth powers, i.e., n6 (e.g., 1, 64, 729, 4096, ). www.vedantu.com 6 11. A digital locker has an alphanumeric (it can have both digits and letters) passcode of length 5. Some example codes are G89P0, 38098, BRJKW, and 003AZ. How many such codes are possible? Answer: 10 digits + 26 letters 36 choices per place. Total codes = 365 = 60,466,176. 12. The worldwide population of sheep (2024) is about 109, and that of goats is also about the same. What is the total population of sheep and goats? (i) 209 (ii) 1011 (iii) 1010 (iv) 1018 (v) 2 109 (vi) 109 + 109 Answer: 109 + 109 = 2 109. Correct option: (v) 2 109. 13. Calculate and write the answer in scientific notation: (i) If each person in the world had 30 pieces of clothing, find the total number of pieces of clothing. (ii) There are about 100 million bee colonies in the world. Find the number of honeybees if each colony has about 50,000 bees. (iii) The human body has about 38 trillion bacterial cells. Find the bacterial population residing in all humans in the world. (iv) Total time spent eating in a lifetime in seconds. Answer: (Taking world population 8 109) (i) 30 8 109 = 2.4 1011 (ii) 108 5 104 = 5 1012 (iii) 3.8 1013 8 109 = 3.04 1023 3.0 1023 (iv) 1.5 hours/day 75 years 1.5 108 seconds www.vedantu.com 7 14. What was the date 1 arab/1 billion seconds ago? Answer: 1 billion seconds 31.7 years. From 27 October 2025 (IST), the date was approximately 17 February 1994. www.vedantu.com 8

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