Solution:
[i]81
Prime factors of 81 = 3\times3\times3\times3
Here one factor 3 is not grouped in triplets.
Therefore 81 must be divided by 3 to make it a perfect cube.
[ii] 128
Prime factors of 128 = 2\times2\times2\times2\times2\times2
Here one factor 2 does not appear in a 3’s group.
Therefore, 128 must be divided by 2 to make it a perfect cube.
[iii] 135
Prime factors of 135 = 3\times3\times3\times5
Here one factor 5 does not appear in a triplet.
Therefore, 135 must be divided by 5 to make it a perfect cube.
[iv] 192
Prime factors of 192 = 2\times2\times2\times2\times2\times3
Here one factor 3 does not appear in a triplet.
Therefore, 192 must be divided by 3 to make it a perfect cube.
[v] 704
Prime factors of 704 = 2\times2\times2\times2\times2\times2\times11
Here one factor 11 does not appear in a triplet.
Therefore, 704 must be divided by 11 to make it a perfect cube.
Step-by-step explanation:
Prime factorising 2916, we get,
2916=2×2×3×3×3×3×3×3
=2
2
×3
6
.
We know, a perfect cube has multiples of 3 as powers of prime factors.
Here, number of 2’s is 2 and number of 3’s is 6.
So we need to multiply another 2 in the factorization to make 2916 a perfect cube.
Hence, the smallest number by which 2916 must be multiplied to obtain a perfect cube is 2.
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1. Perfect Square:
A natural number x is a perfect square if there exists a natural number y such that x=y2. In other words, a natural number x is a perfect square, if it is equal to the product of a number with itself.
2. Properties of Squares Numbers:
[i] A number ending in 2, 3, 7, or 8 is never a perfect square.
[ii] The number of zeroes in the end of a perfect square is never odd. So, a number ending in an odd number of zeroes is never a perfect square.
[iii] Squares of even numbers are always even.
[iv] Squares of odd numbers are always odd.
3. General Properties of Perfect Squares:
[i] For any natural number n, we have n2= [Sum of first n odd natural numbers]
[ii] The square of a natural number, other than 1, is either a multiple of 3 or exceeds a multiple of 3 by 1 .
[iii] The square of a natural number, other than 1, is either a multiple of 4 or exceeds a multiple of 4 by 1.
[iv] There are no natural numbers p and q such that p2=2q2
4. Pythagorean Triplets:
For any natural number n greater than 1, [2n, n2−1, n2+1], is a Pythagorean triplet.
5. Square roots:
The square root of a given natural number n is that natural number which when multiplied by itself gives n as the product and we denote the square root of n by n. Thus, n=m⇔n=m2.
6. Finding Square Roots:
[i] In order to find the square root of a perfect square, resolve it into prime factors; make pairs of similar factors and take the product of prime factors, choosing one out of every pair.
[ii] For finding the square root of a decimal fraction, make the even number of decimal places by affixing a zero, if necessary; mark off periods and extract the square root; putting the decimal point in the square root as soon as the integral part is exhausted.
7. Properties of Square Roots:
For positive numbers a and b, we have
[i] ab=a×b
[ii] ab =ab