8. Find the smallest number by which 2560 must be multiplied so that the product is a
perfect cube.
GIVE ME FULL EXPLANATION.
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Answer:
2560×25 = 64000. which is a perfect cube. hence the smallest no by which 2560 must be multiplied so that product is perfect cube = 5×5=25.
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If you break down 2560 into product of prime factors then 2560 = 2×2×2×2×2×2×2×2×2×5 i.e (2^9)×5.
Now , any perfect cube number is of the form , say if N is a perfect cube number then N = (a^3z)×(b^3y)×(c^3x)…. , where a,b,c represent prime factors of N.
In the above case the power to which 2 is raised is a multiple of 3 but the power of 5 isn't. So the smallest number by which 2560 should be multiplied so that the product is a perfect cube is 5*5.
The resultant number is 2^9 × 5^3 = 64000 which is the cube of 40.
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