Value Comparison with Variable a: Finding the Largest When a>1

We can observe in all the given options that there are fractions where both the numerator and the denominator have terms with identical bases. Therefore we will apply the division law between terms with identical bases in order to solve the exercise:

$\frac{c^m}{c^n}=c^{m-n}$

Proceed to apply it to the given problem. Begin by simplifying each of the suggested options using the above law (options in order):

$\frac{a^4}{a^{-4}}=a^{4-(-4)}=a^{4+4}=a^8$$\frac{a^{10}}{a^9}=a^{10-9}=a^1=a$$\frac{a^4}{a^{-1}}=a^{4-(-1)}=a^{4+1}=a^5$$\frac{a^6}{a^7}=a^{6-7}=a^{-1}$

Remember that any number when raised to the power of 1 equals the number itself, as shown below:

$b^1=b$

Returning to our problem, given that:

$a>1$

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 that option:$a^8$ is correct,

Therefore answer A is correct.