Finding the Largest Value When a > 1: Inequality Comparison

Which represents the the largest?

value given that $a >1$

Notice that in almost all options there are fractions where both 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, let's simplify each of the given options using the above law (options in order):

$\frac{a^9}{a^8}=a^{9-8}=a^1$$\frac{a^{-2}}{a^{10}}=a^{-2-10}=a^{-12}$$a^2$$\frac{a^{20}}{a^{30}}=a^{20-30}=a^{-10}$Let's return 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),

Which means the option:

$a^2$above is correct, it is option C,

Therefore answer C is correct.