Evaluate 9⁴ × 3⁻⁸ × (1/3): Complete Exponent Simplification

First let's note that the number 9 is a power of the number 3:

$9=3^2$

Therefore we can immediately move to a unified base in the problem, in addition we'll recall the law of powers for negative exponents but in the opposite direction:

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

Let's apply this to the problem:

$9^4\cdot3^{-8}\cdot\frac{1}{3}=(3^2)^4\cdot3^{-8}\cdot3^{-1}$

In the first term of the multiplication we replaced the number 9 with a power of 3, according to the relationship mentioned earlier, and simultaneously the third term in the multiplication we expressed as a term with a negative exponent according to the aforementioned law of exponents.

Now let's recall two additional laws of exponents:

a. The law of exponents for power of a power:

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

b. The law of exponents for multiplication between terms with equal bases:

$a^m\cdot a^n=a^{m+n}$

Let's apply these two laws to the expression we got in the last stage:

$(3^2)^4\cdot3^{-8}\cdot3^{-1}=3^{2\cdot4}\cdot3^{-8}\cdot3^{-1}=3^8\cdot3^{-8}\cdot3^{-1}=3^{8+(-8)+(-1)}=3^{8-8-1}=3^{-1}$

In the first stage we applied the law of exponents for power of a power mentioned in a', in the next stage we applied the law of exponents for multiplication of terms with identical bases mentioned in b', then we simplified the resulting expression.

Let's summarize the solution steps:

$9^4\cdot3^{-8}\cdot\frac{1}{3}=(3^2)^4\cdot3^{-8}\cdot3^{-1} =3^{8+(-8)+(-1)}=3^{8-8-1}=3^{-1}$

Therefore the correct answer is answer b'.