Finding the Largest Value: Comparing Numbers with Constraint a>1

Note that in almost all options there are fractions where both the numerator and denominator have identical bases, therefore we will use the division law between terms with identical bases to solve the exercise:

$\frac{c^m}{c^n}=c^{m-n}$Let's apply this to the problem, first we'll simplify each of the given options using the above law (options in order):

$\frac{a^{12}}{a^{10}}=a^{12-10}=a^2$$\frac{a^4}{a^2}=a^{4-2}=a^2$$\frac{a^{-7}}{a^5}=a^{-7-5}=a^{-12}$$a^3$Back to the problem, given that:

$a>1$therefore the option with the largest value will be the one where $a$has the largest exponent (for emphasis - a positive exponent is greater than a negative exponent),

meaning the option:

$a^3$above is correct,

therefore answer D is correct.