Question

check wether the square is perfect square or not i 729 in prime factorisation method

Answer 1

Answer: NO, 9 is not in the list of numbers that are never perfect squares. Let's continue to the next step.

Step 2:

We now need to obtain the digital root of the number. Here's how you do it:

Split the number up and add each digit together:

7 + 2 + 9 = 18

If the answer is more than one digit, you would add each digit of the answer together again:

1 + 8 = 9

What is the digital root of number 729?

Answer: 9

Step 3:

So now we know the digital root of 729 is 9. Is 9 in the list of digital roots that are always a square root (1, 4, 7 or 9)?

Answer: YES, 9 is in the list of digital roots that are always perfect squares. We can conclude that 729 could be a perfect square!

Factoring

OK, so now we know that 729 could be a perfect square. We have to find the factors of the number to be sure.

Here are all of the factors of 729:

1 x 7293 x 2439 x 8127 x 27  

Highlighted in orange above is the factor combination that makes 729 a perfect square. Do you see why? A number can only be a perfect square if the product of two exactly the same numbers is equal to the original number.  

Here's the proof: 27 x 27 = 729  

Answer 2

Hey there !

Here's the answer !

Prime Factorisation of 729 = 3 × 3 × 3 × 3 × 3 × 3 = 3⁶

Since the factors are of powers in terms of of even powers, this number can be considered as a perfect square.

3⁶ = 3³ × 3³

=> √ ( 3⁶ ) = √ ( 3³ × 3³  ) = 3³

Hence the number 729 is a perfect square which can be written as ( 3³  )² or 27².

Hope my answer helped !