Question

find the smallest number by which 432 must be multiplied so that the product becomes a perfect cube

Answer 1

Answer:

3

Step-by-step explanation:

by prime factorization  

we get 2^3*3^3

as we see that 3 is not a triplet, so we should

432 *3=1296

so the smallest number is =3

2    432

2      216

2        18

3         9

3          3

Answer 2

A perfect cube of a number is a number that is equal to the number, multiplied by itself, three times. If x is a perfect cube of y, then $x = y^3$. Therefore, if we take the cube root of a perfect cube, we get a natural number and not a fraction.

Prime factorization of $432=2\times 2\times 2\times 2\times3\times 3\times 3$

$=2^4\times 3^3$

If we multiply the number.

$2$ twice,

i.e.,

$=2\times 2 \times 2\times 2$

$=2^4$

$=2\times 2\times 2\time 2\times\ 2\times2\times 2$

$=2^6$

$2^4\times 3^3\times 2^2=432\times 2^2$

$=432\times 4$

$=1728$

$728=2^6\times 3^3$

$=2^3\times 2^3\times 3^3$      $\therefore$ $a^m\times b^m =(ab)^m$

$=(2\times 2\times 3)^3$

$1728=(12)^3$

Hence, the number should be multiplied by $4$ to make it a perfect cube.