Simplify (-1/8)^8 × (-1/8)^-3: Complete Exponent Operation

Apply the power law for multiplication between terms with identical bases:

$a^m\cdot a^n=a^{m+n}$We'll apply this law to the problem:

$\big(-\frac{1}{8}\big)^8\cdot\big(-\frac{1}{8}\big)^{-3}=\big(-\frac{1}{8}\big)^{8+(-3)}=\big(-\frac{1}{8}\big)^{8-3}=\big(-\frac{1}{8}\big)^5$

In the first stage we applied the above power law and in the following stages we simplified the expression in the exponent,

Let's continue and use the power law for power of terms in parentheses:

$(x\cdot y)^n=x^n\cdot y^n$

We'll apply this law to the expression that we obtained in the last stage:

$\big(-\frac{1}{8}\big)^5=\big(-1\cdot\frac{1}{8}\big)^5=(-1)^5\cdot\big(\frac{1}{8}\big)^5=-1\cdot\big(\frac{1}{8}\big)^5=-\big(\frac{1}{8}\big)^5$

In the first stage we presented the expression in parentheses as a multiplication between negative one and a positive number. In the next stage we applied the above power law and then simplified the expression we obtained whilst noting that negative one to an odd power will (always) give the result negative one.

Next we'll recall two additional power laws:

a. The negative power law:

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

b. The power law for power of a power:

$(a^m)^n=a^{m\cdot n}$

We'll continue and apply these two laws to the expression that we obtained in the last stage:

$-\big(\frac{1}{8}\big)^5=-(8^{-1})^5=-8^{(-1)\cdot5}=-8^{-5}$

In the first stage we presented the fraction inside the parentheses as a term with a negative power using the above power law for negative power mentioned in a. above. In the next stage we applied the power law for power of a power mentioned in b. above carefully, given that the term inside the parentheses has a negative power. We then simplified the expression in the exponent.

Let's summarize the solution :

$\big(-\frac{1}{8}\big)^8\cdot\big(-\frac{1}{8}\big)^{-3}=\big(-\frac{1}{8}\big)^5=-\big(\frac{1}{8}\big)^5=-(8^{-1})^5=-8^{-5}$

Therefore the correct answer is answer d.