4 Mark Question

Question 14 Marks

Which of the following are perfect cubes?
(i) 400 $\qquad~~$ (ii) 3375 $\quad$ (iii) 8000
(iv) 15625 $\quad$ (v) 9000 $\quad$ (vi) 6859
(vii) 2025 $\quad~$ (viii) 10648

Answer

(i) We have, 400
Resolving 400 into prime factors,
we get

$400=\underline{2 \times 2 \times 2} \times 2 \times 5 \times 5$
Clearly, the prime factors 2 and 5 do not appear in group of three (triples).
So, 400 is not a perfect cube.
(ii) We have, 3375
Resolving 3375 into prime factors, we get

3	3375
3	1125
3	375
5	125
5	25
5	5
	1

Clearly, the prime factors of 3375 are grouped into triples of equal factors and no factor is left over.
So, 3375 is a perfect cube.
(iii) 8000 is a perfect cube.
(iv) 15625 is a perfect cube
(v) 9000 is not a perfect cube.
(vi) 6859 is a perfect cube.
(vii) 2025 is not a perfect cube.
(viii) 10648 is a perfect cube.

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Question 24 Marks

Check which of the following are perfect cubes :
16000

Answer

We have, 16000
Resolving 16000 into prime factors,
we get

2	16000
2	8000
2	4000
2	2000
2	1000
2	500
2	250
5	125
5	25
5	5
	1

$16000=\underline{2 \times 2 \times 2} \times \underline{2 \times 2 \times 2} \times 2\times \underline{5 \times 5 \times 5}$
Clearly, the prime factor 2 does not appear in group of three (triples).
So, 16000 is not a perfect cube.

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Question 34 Marks

Find the smallest number by which each of the following number must be multiplied to obtain a perfect cube :
256

Answer

Given number is 256.
Resolving 256 into prime factors, we get

2	256
2	128
2	64
2	32
2	16
2	8
2	4
2	2
	1

$256=\underline{2 \times 2 \times 2} \times \underline{2 \times 2 \times 2} \times 2 \times 2$
The prime factor 2 does not appear in a group of three (triples).
Therefore, 256 is not a perfect cube.
To make it a cube, we need one more prime number 2.
In that case,
$\begin{aligned} 256 \times 2 & =\underline{2 \times 2 \times 2} \times \underline{2 \times 2 \times 2} \times \underline{2 \times 2 \times 2} \\ & =512\end{aligned}$
which is a perfect cube.
Hence, the smallest number by which 256 should be multiplied to make a perfect cube is 2.

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Question 44 Marks

Find the smallest number by which each of the following number must be multiplied to obtain a perfect cube :
243

Answer

Given number is 243.
Resolving 243 into prime factors, we get

3	243
3	81
3	27
3	9
3	3
	1

$243=\underline{3 \times 3 \times 3} \times 3 \times 3$
The prime factor 3 does not appear in a group of three triples.
Therefore, 243 is not a perfect cube.
To make it a cube, we need one more prime number 3.
In that case,
$243 \times 3=\underline{3 \times 3 \times 3} \times \underline{3 \times 3 \times 3}=729$, which is a perfect cube.
Hence, the smallest number by which 243 should be multiplied to make a perfect cube is 3.

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Question 54 Marks

Which of the following numbers are not perfect cubes?
(i) 216$\qquad$(ii) 128$\quad$(iii) 1000
(iv) 100$\quad~~$(v) 46656

Answer

(i) We have, 216
Resolving 216 into prime factors, we get

2	216
2	108
2	54
3	27
3	9
3	3
	1

$216=\underline{2 \times 2 \times 2} \times \underline{3 \times 3 \times 3}$
Clearly, prime factors of 2 and 3 occurs in a group of three (triples).
So, 216 is a perfect cube.
(ii) We have, 128
Resolving 128 into prime factors, we get

2	128
2	64
2	32
2	16
2	8
2	4
2	2
	1

$128=\underline{2 \times 2 \times 2} \times \underline{2 \times 2 \times 2} \times 2$
Clearly, the prime factor of 2 does not appear in group of three (triples).
So, 128 is not a perfect cube.
(iii) 1000 is a perfect cube.
(iv) 100 is not a perfect cube.
(v) 46656 is a perfect cube.

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