Applying Combined Exponents Rules: Presenting powers in the denominator as powers with negative exponents

$\frac{1}{12^3}=\text{?}$

To begin with, we must remind ourselves of the Negative Exponent rule:

$a^{-n}=\frac{1}{a^n}$We apply it to the given expression :

$\frac{1}{12^3}=12^{-3}$Therefore, the correct answer is option A.

$12^{-3}$

$\frac{1}{2^9}=\text{?}$

We use the power property for a negative exponent:

$a^{-n}=\frac{1}{a^n}$We apply it to the given expression:

$\frac{1}{2^9}=2^{-9}$

Therefore, the correct answer is option A.

$2^{-9}$

$\frac{1}{8^3}=\text{?}$

We use the negative exponent rule.

$b^{-n}=\frac{1}{b^n}$

We apply it to the problem in the opposite sense.:

$$$$$\frac{1}{8^3}=8^{-3}$

Therefore, the correct answer is option A.

$8^{-3}$

$\frac{1}{(-2)^7}=?$

To begin with we deal with the expression in the denominator of the fraction. Making note of the power rule for exponents (raising an exponent to another exponent):

$(a^m)^n=a^{m\cdot n}$We obtain the following:

$(-2)^7=(-1\cdot2)^7=(-1)^7\cdot2^7=-1\cdot2^7=-2^7$

We then return to the initial problem and apply the above information:

$\frac{1}{(-2)^7}=\frac{1}{-2^7}=\frac{1}{-1}\cdot\frac{1}{2^7}=-\frac{1}{2^7}$$$

In the last step we remember that:

$\frac{1}{-1}=-1$

Next, we remember the Negative Exponent rule ( raising exponents to a negative power)

$a^{-n}=\frac{1}{a^n}$We apply it to the expression we obtained in the last step:

$-\frac{1}{2^7}=-2^{-7}$Let's summarize the steps of the solution:

$\frac{1}{(-2)^7}=-\frac{1}{2^7} = -2^{-7}$

$$$$Therefore, the correct answer is option C.

$(-2)^{-7}$

$\frac{27}{3^8}=\text{?}$

$3^{-5}$

$\frac{1}{a^n}=\text{?}$

$a\ne0$

$a^{-n}$$$

$\frac{2}{4^{-2}}=\text{?}$

$2\cdot4^2$

$\frac{10}{(-5)^3}=\text{?}$

$-2(-5)^{-2}$