Let's simplify each of the given expressions step-by-step:
### 1. [tex]\(\frac{5^0 \cdot 5^3}{5^2}\)[/tex]
Using the properties of exponents, specifically:
[tex]\[
\frac{a^m \cdot a^n}{a^p} = a^{m+n-p}
\][/tex]
First, apply the exponent rules to the numerator:
[tex]\[
5^0 \cdot 5^3 = 5^{0+3} = 5^3
\][/tex]
Now consider the whole expression:
[tex]\[
\frac{5^3}{5^2} = 5^{3-2} = 5^1 = 5
\][/tex]
So, the expression simplifies to:
[tex]\[
\boxed{5}
\][/tex]
### 2. [tex]\((a^2)^4 \cdot c^3 \cdot c^{-2}\)[/tex]
Using the property of exponents that states [tex]\((a^m)^n = a^{m \cdot n}\)[/tex] and [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex]:
First, simplify [tex]\((a^2)^4\)[/tex]:
[tex]\[
(a^2)^4 = a^{2 \cdot 4} = a^8
\][/tex]
Next, simplify [tex]\(c^3 \cdot c^{-2}\)[/tex]:
[tex]\[
c^3 \cdot c^{-2} = c^{3-2} = c^1 = c
\][/tex]
Combine the simplified expressions:
[tex]\[
a^8 \cdot c
\][/tex]
So, the expression simplifies to:
[tex]\[
\boxed{a^8 \cdot c}
\][/tex]
### 3. [tex]\(\frac{P^3 \cdot (Q^1)^2}{P^2}\)[/tex]
Using the properties of exponents, specifically [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]:
First, simplify [tex]\((Q^1)^2\)[/tex]:
[tex]\[
(Q^1)^2 = Q^{1 \cdot 2} = Q^2
\][/tex]
Now, consider the entire expression with [tex]\(P^3\)[/tex] and [tex]\(P^2\)[/tex]:
[tex]\[
\frac{P^3 \cdot Q^2}{P^2} = \frac{P^{3} \cdot Q^2}{P^2} = P^{3-2} \cdot Q^2 = P^1 \cdot Q^2 = P \cdot Q^2
\][/tex]
So, the expression simplifies to:
[tex]\[
\boxed{P \cdot Q^2}
\][/tex]
