Fully simplify and write your answer in exponential form with positive exponents only. Example: 153\frac{1}{5^3}

Question

5
3

1
​
not 5−35^{-3}5
−3
. Click Here for Help Video.

1z8⋅1z4=\frac{1}{z^{8}}\cdot\frac{1}{z^{4}}=
z
8

1
​
⋅
z
4

1
​
=

Answer 1

To simplify the expression \(\frac{1}{z^{8}} \cdot \frac{1}{z^{4}}\), follow these steps:

1. Rewrite the expression by combining the fractions: \[ \frac{1}{z^{8}} \cdot \frac{1}{z^{4}} = \frac{1 \cdot 1}{z^{8} \cdot z^{4}} = \frac{1}{z^{8 + 4}} = \frac{1}{z^{12}} \]

2. Since the goal is to express the result in exponential form with positive exponents, you can write: \[ \frac{1}{z^{12}} = z^{-12} \]

However, since you've asked for a positive exponent, we can rewrite it as: \[ z^{12} = z^{-12} \] But in this case, we realize we have been given instructions not to express with negative exponents, so we will retain it as:

\[ \frac{1}{z^{12}} \text{ or } z^{-12} \]

Since the expression was to be expressed positively, and you already have the term as \(\frac{1}{z^{12}}\) is indeed indicating a positive representation in terms of \(z^{-12}\).

Final Answer: \[ \frac{1}{z^{12}} \]