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To use prime factorization to find a square root, after factorizing the number into its primes split each prime into two groups so that the same number (half the number) are in each group - the square root is the product of the primes in one of the groups (both groups are equal). If any prime occurs an odd number of times, then the square root is irrational and to leave the answer in surd form, collect the extra primes into a separate group and multiply the product of one of the equal groups by the square root of the product of the primes in this extra group:

1728 = 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 3

= (2 x 2 x 2) x (2 x 2 x 2) x (3) x (3) x 3

= (2 x 2 x 2 x 3) x (2 x 2 x 2 x 3) x 3 = 24 × 24 × 3

→ √1728 = 24 × √3

Q: What is the square root of 1728 using prime factorization?

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26 x 33 = 1728

As a product of its prime factors in exponents: 2^6 times 3^3 = 1728

1728 square feet

144 square feet

12 square feet

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26 x 33 = 1728

1728 = 2 * 2 * 2 * 2 * 2 * 2 * 3 * 3 * 3 = 2^6 * 3^3 They are in exponents: 2^6 and 3^3

The prime factors of 1728 are 2 and 3

As a product of its prime factors in exponents: 2^6 times 3^3 = 1728

1728 square feet

144 square feet

12 square feet

You cannot because it is not a perfect square.

36'x48'=1728 square feet.

24√3 or 41.5692194 or 41.57, to the justified number of significant digits since 1728 has only four.

1728 = 2 x 864 = 2x2 x 432 = 2x2x2 x 216 = 2x2x2 x 2 x 108 = 2x2x2 x 2x2 x 54 = 2x2x2 x 2x2x2 x 27 = 2x2x2 x 2x2x2 x 3 x 9 = 2x2x2 x 2x2x2 x 3x3x3 = 2x2x3 x 2x2x3 x 2x2x3 = 12 x 12 x 12 cube root = 12

Out of the factor pairs of 1728, only 144 and 12 have a square/square root relationship. 12 has to be the cube root of 1728. (1728,1)(864,2)(576,3)(432,4)(288,6)(216,8)(192,9)(144,12)(108,16)(96,18)(72,24)(64,27)(54,32)(48,36)