Write the answer to this calculation as the product of powers of prime numbers:
$$4^3\times9^3\times20^2$$
I've browsed a few related questions suggested and I don't think the ones I have found are as basic as what I'm asking. I'm doing an online maths course to get back my long forgotten A-Level maths and I have came across this question;
I am happy to work out the value of the given calculation and then factorise by table until I find the answer, I'm comfortable with getting the right answer doing it that way. My question was more, is there a more elegant way, that the course is expecting, that I am missing?
I've stared at it for a while trying to see through any obvious relationships in the numbers and came up with nothing though my maths brain has been dormant for a decade. I'm not looking for a solution, but more a suggestion that 'yes, there is a cleaner way than just working out the big number and factorising down'. If there ISN'T then great, I know what to do 🙂
I'm sure there are various equations that tackle this, though bear in mind I am taking a first year A-Level course so I'm more curious that I am missing something fundamental..
Thanks for your time on this unbelievably easy question for most of you guys!
Best Answer
We will use the rules $a^k \times a^j = a^{k + j}$ and $(a^k)^j=a^{kj}$ repeatedly to simplify. The first step is to break each term into primes then simplify and rearrange them to get it into the final form. The calcluation goes as follows: $$4^3 \times 9^3 \times 20^2=(2^2)^3 \times (3^2)^3 \times (2^2 \times 5)^2=2^6 \times 3^6 \times 2^4 \times 5^2 = 2^{6 + 4}\times 3^6 \times 5^2=2^{10}\times 3^6 \times 5^2$$