Power of a Quotient Rule for Exponents: Identify the greater value

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.

Note that in all options there are fractions where both numerator and denominator have terms with 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 it to the problem, first we'll simplify each of the suggested options using the above law (options in order):

$\frac{a^{17}}{a^{20}}=a^{17-20}=a^{-3}$$\frac{a^{10}}{a^1}=a^{10-1}=a^9$$\frac{a^3}{a^{-2}}=a^{3-(-2)}=a^{3+2}=a^5$$\frac{a^2}{a^4}=a^{2-4}=a^{-2}$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),

meaning the option:

$a^9$above is correct, it came from option B in the answers,

therefore answer B is correct.

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.

Note that in all options there are fractions where both numerator and denominator have terms with 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 it to the problem, first we'll simplify 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}$

where we also used the fact that any number to the power of 1 equals the number itself, meaning that:

$b^1=b$

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:$a^8$above is correct, it came from option A in the answers,

therefore answer A is correct.