Power Play | Class 8 Mathematics Notes

To find the units digit of 2224 ÷ 432, first note that 4 = 22. This implies 4 is 2 squared. The question likely means to find the units digit of 2 raised to the power 224 divided by 4 raised to the power 32, or possibly 2^224 divided by 4^32. Since 4 = 2^2, 4^32 = (2^2)^32 = 2^(64). So the expression is 2^224 ÷ 2^64 = 2^(224-64) = 2^160. The units digit of powers of 2 cycle every 4 powers: 2^1=2, 2^2=4, 2^3=8, 2^4=6, then repeats. 160 mod 4 = 0, so units digit is 6.

Each container has 5 bottles. Every day, one new container is added. After 40 days, number of containers = 40. Total bottles = 40 × 5 = 200 bottles.

For each number, express as product of powers in three different ways:

(i) 643

• 64 = 8^2 = 4^3 = 2^6
• So possible products:

1) 8^2 2) 4^3 3) 2^6

(ii) 1928

• 1928 = 19 × 28
• 19 is prime, 28 = 2^2 × 7
• Possible products:

1) 19 × 28 2) (19 × 2)^2 × 7^{-1} (not integer power, so avoid) 3) 19^1 × (2^2 × 7)^1

(iii) 32^{-5}

• 32 = 2^5
• So 32^{-5} = (2^5)^{-5} = 2^{-25}
• Possible products:

1) 2^{-25} 2) (2^{-5})^{5} 3) (2^{-1})^{25}

(i) Cube numbers are also square numbers.

• Only sometimes true. For example, 64 = 4^3 = 8^2 is both cube and square, but 8 = 2^3 is not a square.

(ii) Fourth powers are also square numbers.

• Always true. Because (n^2)^2 = n^4, so fourth powers are squares of squares.

(iii) The fifth power of a number is divisible by the cube of that number.

• Always true. Since n^5 = n^3 × n^2, n^5 is divisible by n^3.

(iv) The product of two cube numbers is a cube number.

• Always true. Since a^3 × b^3 = (