Sure, let's solve each part of the problem step-by-step:
### Step 1: Simplifying [tex]\(\frac{5^0 \cdot 5^3}{5^2}\)[/tex]
First, we need to understand the properties of exponents. Specifically:
[tex]\[ a^m \cdot a^n = a^{m+n} \][/tex]
[tex]\[ \frac{a^m}{a^n} = a^{m-n} \][/tex]
[tex]\[ a^0 = 1 \][/tex]
Using these properties, we can simplify [tex]\(\frac{5^0 \cdot 5^3}{5^2}\)[/tex]:
1. [tex]\(5^0 = 1\)[/tex]
2. Substitute this into the fraction:
[tex]\[
\frac{1 \cdot 5^3}{5^2}
\][/tex]
3. Simplify the exponents in the numerator and denominator:
[tex]\[
\frac{5^3}{5^2} = 5^{3-2} = 5^1 = 5
\][/tex]
Therefore,
[tex]\[
\frac{5^0 \cdot 5^3}{5^2} = 5
\][/tex]
### Step 2: Simplifying [tex]\(\left(a^2\right)^4 \cdot c^3 \cdot c^{-2}\)[/tex]
For this type of problem, we use the properties:
[tex]\[ (a^m)^n = a^{mn} \][/tex]
[tex]\[ a^m \cdot a^n = a^{m+n} \][/tex]
First, simplify [tex]\((a^2)^4\)[/tex]:
1.
[tex]\[
(a^2)^4 = a^{2 \cdot 4} = a^8
\][/tex]
Next, simplify [tex]\(c^3 \cdot c^{-2}\)[/tex]:
2.
[tex]\[
c^3 \cdot c^{-2} = c^{3 + (-2)} = c^1 = c
\][/tex]
Therefore, combining the results,
[tex]\[
(a^2)^4 \cdot c^3 \cdot c^{-2} = a^8 \cdot c
\][/tex]
### Step 3: Simplifying [tex]\(\frac{P^3 \cdot (Q^1)^2}{P^2}\)[/tex]
Here we use the properties again:
[tex]\[ a^m \cdot a^n = a^{m+n} \][/tex]
[tex]\[ \frac{a^m}{a^n} = a^{m-n} \][/tex]
[tex]\[ (a^m)^n = a^{mn} \][/tex]
First, simplify [tex]\((Q^1)^2\)[/tex]:
1.
[tex]\[
(Q^1)^2 = Q^{1 \cdot 2} = Q^2
\][/tex]
Next, substitute into the expression:
2.
[tex]\[
\frac{P^3 \cdot Q^2}{P^2}
\][/tex]
Now, simplify the exponents:
3.
[tex]\[
\frac{P^3}{P^2} = P^{3-2} = P^1 = P
\][/tex]
Finally, combine the results:
[tex]\[
P \cdot Q^2 = P^1 \cdot Q^2
\][/tex]
Therefore,
[tex]\[
\frac{P^3 \cdot (Q^1)^2}{P^2} = P \cdot Q^2
\][/tex]
### Final Results
[tex]\[
\frac{5^0 \cdot 5^3}{5^2} = 5, \quad (a^2)^4 \cdot c^3 \cdot c^{-2} = a^8 \cdot c, \quad \frac{P^3 \cdot (Q^1)^2}{P^2} = P \cdot Q^2
\][/tex]
[tex]\[
\boxed{5, \ a^8 \cdot c, \ P \cdot Q^2}
\][/tex]
