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Q. State true or false:

for any integer $\inline m,m^{2}< m^{3}.$ Why?

The detailed solution for the above-written question is as follows. False.                   or            or           or          Now put any number less than 1, we see that this relation doesn't hold. So for m<1 this condition is not true.

Q. Check which of the following are perfect cubes.

(i) 2700

(ii) 16000

(iii) 64000

(iv) 900

(v) 125000

(vi) 36000

(vii) 21600

(viii) 10,000

(ix) 27000000

(x) 1000.

What pattern do you observe in these perfect cubes?

The detailed solution for the above-written question is as follows By prime factorization: (i)  .      So it is not a perfect cube. (ii)  .   So it is not a perfect cube. (iii)  .  So it is a perfect cube. (iv)  .   So it is not a perfect cube. (v)  .   So it is a perfect cube. (vi) .  So it is not a perfect cube. (vii) .  So it is not a perfect cube. (viii) .  So it is not a perfect...

Q. Which of the following are perfect cubes?

1. 400

2. 3375

3. 8000

4. 15625

5. 9000

6. 6859

7. 2025

8. 10648

We will find it by prime factorisation whether they make pair of three prime numbers or not.

(1)  $400 = 2\times2\times2\times2\times5\times5$ .   So not a perfect cube.

(2)  $3375 = 3\times3\times3\times5\times5\times5$.  So it is a perfect cube.

(3)  $8000 = 2\times2\times2\times2\times2\times2\times5\times5\times5$.  So it is a perfect cube.

(4)  $15625 = 5\times5\times5\times5\times5\times5$.    So it is a perfect cube.

(5)  $9000 = 2\times2\times2\times3\times3\times5\times5\times5$.  So it is not a perfect cube.

(6)  $9000 = 19\times19\times19$.   So it is a perfect cube.

(7)  $2025 = 3\times3\times3\times3\times5\times5$.   So it is not a perfect cube.

(8)  $10648 = 2\times2\times2\times11\times11\times11$.  So it is a perfect cube.

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Q. Consider the following pattern.

$2^{3}-1^{3}=1+2\times 1\times 3$

$3^{3}-2^{3}=1+3\times2\times 3$

$4^{3}-3^{3}=1+4\times 3\times 3$

Using the above pattern, find the value of the following.

(iv)  $51^{3}-50^{3}$

The detailed solution for the above written question is mentioned below

Q. Consider the following pattern.

$2^{3}-1^{3}=1+2\times 1\times 3$

$3^{3}-2^{3}=1+3\times2\times 3$

$4^{3}-3^{3}=1+4\times 3\times 3$

Using the above pattern,find the value of the following.

(iii)  $20^{3}-19^{3}$

The detailed solution for the above-written question is mentioned below,

The detailed solution for all the above-written question is as follows

Using the above pattern,

$2^{3}-1^{3}=1+2\times 1\times 3$

$3^{3}-2^{3}=1+3\times2\times 3$

$4^{3}-3^{3}=1+4\times 3\times 3$

find the value of the following.

(ii) $12^{3}-11^{3}$

The detailed solution of the above-written question is as follows,

Using the above pattern,

$2^{3}-1^{3}=1+2\times 1\times 3$

$3^{3}-2^{3}=1+3\times2\times 3$

$4^{3}-3^{3}=1+4\times 3\times 3$

find the value of the following.

(i)  $7^{3}-6^{3}$

The value of the following question is:

Q.Express the following numbers as the sum of odd numbers using the above pattern?

(c)  $7^{3}$

The detailed solution for the above-mentioned question is as follows 73 = 43 + 45 + 47 + 49 + 51 + 53 + 55

Q. Express the following numbers as the sum of odd numbers using the above pattern?

(b)  $8^{3}$

The detailed solution for the above-mentioned question is as follows     512  =>  57 + 59 + 61 + 63 + 65 + 67 + 69 + 71

Q. Express the following numbers as the sum of odd numbers using the above pattern?

(a)  $6^{3}$

The detailed solution for the above mentioned question is as follows,   216  =>  31 + 33 + 35 + 37 + 39 + 41

Q. Find the one’s digit of the cube of each of the following numbers.

(viii) 53

The detailed solution for the above-mentioned question is as follows, Since the given number has 3 at units place, so, its cube will end with 7.

Q. Find the one’s digit of the cube of each of the following numbers.

(vii) 5022

Q. Find the one’s digit of the cube of each of the following numbers.

(vi) 77

The detailed solution for above-mentioned question is as follows, The given number is ending with 7, so its cube will end with 3.

Q. Find the one’s digit of the cube of each of the following numbers.

(v) 1024

The solution to the above-mentioned question is as follows, The given digit is ending with 4. So the one’s digit of the cube of 1024 will be 4.

Q. Find the one’s digit of the cube of each of the following numbers.

(iv) 1005

The detailed solution for the above-mentioned questions is as follows Since the given number ends with 5, so one's digit of its cube will also end with 5.

Q. Find the one's digit of the cube of each of the following numbers.

(iii) 149

The detailed solution for the above-mentioned question is as follows, Since the given number has 9 at units place,  so the one’s digit of the cube of  149 will be 9.

Q. Find the one’s digit of the cube of each of the following numbers.

(ii) 8888

The detailed solution for the above-mentioned question is as follows Since the given number ends with 8, so the one’s digit of the cube of 8888 will be 2.

Q. Find the one’s digit of the cube of each of the following numbers.

(i) 3331

The detailed solution for the above-mentioned question is as follows, Since the given number ends with 1, so the one’s digit of the cube of 3331 will be 1.
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