Certainly! Let's solve the problem step-by-step using the information provided.
We are given the expression for \( P(x, y) \) as:
[tex]\[ P(x, y)=a^2 x^{a^{a-b}} \][/tex]
### Step 1: Assign Values to Variables
You need to assign values to the variables \( a \), \( b \), \( x \), and \( y \). Based on the provided information:
[tex]\[ a = 0.5 \][/tex]
[tex]\[ b = 0.2 \][/tex]
[tex]\[ x = 4 \][/tex]
[tex]\[ y = 3 \][/tex]
### Step 2: Calculate the Exponent
The exponent is calculated as follows:
[tex]\[ \text{Exponent} = a^{a-b} \][/tex]
We need to find the value of \( a - b \):
[tex]\[ a - b = 0.5 - 0.2 = 0.3 \][/tex]
Now, we raise \( a \) to the power of \( a - b \):
[tex]\[ a^{a-b} = 0.5^{0.3} \][/tex]
From the numerical result provided, we know:
[tex]\[ 0.5^{0.3} \approx 0.8122523963562356 \][/tex]
### Step 3: Calculate \( P(x, y) \)
We can now use the computed exponent to find \( P(x, y) \):
[tex]\[ P(x, y) = a^2 \times x^{a^{a-b}} \][/tex]
First, compute \( a^2 \):
[tex]\[ a^2 = (0.5)^2 = 0.25 \][/tex]
Next, compute \( x^{a^{a-b}} \):
[tex]\[ x^{a^{a-b}} = 4^{0.8122523963562356} \][/tex]
From the numerical result provided, we know:
[tex]\[ 4^{0.8122523963562356} \approx 3.0833631000341936 \][/tex]
Multiply \( a^2 \) by \( x^{a^{a-b}} \):
[tex]\[ P(x, y) = 0.25 \times 3.0833631000341936 \approx 0.7708407750085484 \][/tex]
### Conclusions with Final Values
Thus, substituting the values, we find:
[tex]\[ a = 0.5 \][/tex]
[tex]\[ b = 0.2 \][/tex]
[tex]\[ x = 4 \][/tex]
[tex]\[ y = 3 \][/tex]
[tex]\[ a^{a-b} \approx 0.8122523963562356 \][/tex]
[tex]\[ P(x, y) \approx 0.7708407750085484 \][/tex]
So, the final values are:
- \( a = 0.5 \)
- \( b = 0.2 \)
- \( x = 4 \)
- \( y = 3 \)
- Exponent \( a^{a-b} \approx 0.8122523963562356 \)
- [tex]\( P(x, y) \approx 0.7708407750085484 \)[/tex]
