Vtyjkyuk

I am studying about powers for a discipline in college and the teacher asked me to simplify the following expression to transform it into the form of a single power,

$$
(16^2 times 64^3)div1024^2
$$

I can simplify to,

$$
2^6
$$

But, take many steps to get this result,

$$
(16^2 times 64^3)div1024^2 \ implies(16times16)times(64times64times64)div(1024times1024) \ implies 256 times262144div1048576\ implies67108864div1048576=64\ 64implies2^6 \ (16^2 times 64^3)div1024^2 implies 2^6
$$

However I would like to know if there is a shorter or simpler way to simplify expression $(16^2 times 64^3)div1024^2$ ?

beginalign
& (16^2 times 64^3)div1024^2 \[10pt]
= & (2^4)^2 times (2^6)^3 div (2^10)^2 \[10pt]
= & 2^8 times 2^18 div 2^20 \[10pt]
= & 2^8+18-20.
endalign

By writing these out as powers of primes (namely $2$), we have
$$
frac16^2 times 64^31024^2 = frac(2^4)^2 times (2^6)^3(2^10)^2 = frac2^8times2^182^20 = frac2^262^20 = 2^6.
$$

Too complicated. Notice that:

• $16 = 2^4$;


• $64 = 2^6$;


• $1024 = 2^10$.

Therefore:

$$beginarray[rcl]
((16^2 times 64^3)div 1024^2 & = & (2^8 times 2^18)div 2^20 \
& = & 2^26div 2^20 = 2^6 = 64. \
endarray$$

Notice that all can be represented in powers of 2

$$16^2=(2^4)^2=2^8$$
$$64^3=(2^6)^3=2^18$$
$$1024^2=(2^10)^2=2^20$$
$$frac2^8cdot 2^182^20=frac2^262^20=2^6$$

$$(16^2 times 64^3)div1024^2=2^8 times 2^18 /2^20 =2^6=64$$