Question

How do you express 726 as a product of its prime factors in index form?

Answer 1

Hint: To express a number in its product form of prime factors, we have to express it as the product of prime factors which are prime. a factor can appear more than once in its product form. As all factors are prime the product form should contain prime numbers like 2, 3, 5, 7, 11, etc. which are prime. To express the product form in index form we have to check the factors which are more than once in the product form.

Complete answer:
The given number is 726. The smallest prime factor it has is 2. Hence, we can express 726 as,
\[\Rightarrow 726=2\times 363\]
The smallest prime factor 363 has is 3. Hence, we can express the above product for as,
\[\Rightarrow 2\times 363=2\times 3\times 121\]
The smallest prime factor 121 has is 11. Hence, the above product form can be expressed as,
\[\Rightarrow 2\times 3\times 121=2\times 3\times 11\times 11\]
The numbers in the product form we got are 2, 3, and 11. All of them are prime and can not be factored further hence, the prime factors of 726 are 2, 3, and 11.
So, the prime factor product form of 726 is \[2\times 3\times 11\times 11\].
As we can see that in this form 11 is appearing twice.
So, the product form can be written in index form as \[2\times 3\times {{11}^{2}}\].
726 can be expressed as \[2\times 3\times {{11}^{2}}\], a product of prime factors in index form.

Note: If more than one number is appearing more than once in the prime factor product form of the given number. We have to do the same thing for each of them. It should be remembered that expressing in product form and finding the factors are different things.