Sure, let's solve the expression step-by-step.
We are given the expression:
[tex]\[ \frac{1}{1 + x^{a+b}} + \frac{1}{1 + x^{b-a}} \][/tex]
Let's use the values [tex]\( a = 2 \)[/tex], [tex]\( b = 3 \)[/tex], and [tex]\( x = 5 \)[/tex].
### Step 1: Calculate [tex]\(x^{a+b}\)[/tex]
First, we need to calculate the exponent [tex]\(a + b\)[/tex]:
[tex]\[ a + b = 2 + 3 = 5 \][/tex]
Now, we calculate [tex]\(x^{a+b}\)[/tex]:
[tex]\[ x^{a+b} = 5^5 = 3125 \][/tex]
### Step 2: Calculate the first term
We will plug this result into the first term:
[tex]\[ \frac{1}{1 + x^{a+b}} = \frac{1}{1 + 3125} = \frac{1}{3126} \][/tex]
### Step 3: Calculate [tex]\(x^{b-a}\)[/tex]
Next, we need to calculate the exponent [tex]\(b - a\)[/tex]:
[tex]\[ b - a = 3 - 2 = 1 \][/tex]
Now, we calculate [tex]\(x^{b-a}\)[/tex]:
[tex]\[ x^{b-a} = 5^1 = 5 \][/tex]
### Step 4: Calculate the second term
We will plug this result into the second term:
[tex]\[ \frac{1}{1 + x^{b-a}} = \frac{1}{1 + 5} = \frac{1}{6} \][/tex]
### Step 5: Sum the results
Now let's sum the two results we obtained:
[tex]\[ \frac{1}{3126} + \frac{1}{6} \][/tex]
Converting these into decimals for easier addition:
[tex]\[ \frac{1}{3126} \approx 0.0003198976327575176 \][/tex]
[tex]\[ \frac{1}{6} \approx 0.16666666666666666 \][/tex]
Finally, summing these two values:
[tex]\[ 0.0003198976327575176 + 0.16666666666666666 = 0.16698656429942418 \][/tex]
### Result:
The value of the given expression is:
[tex]\[ 0.16698656429942418 \][/tex]
Additionally, breaking it down we get:
[tex]\[ \frac{1}{3126} \approx 0.0003198976327575176 \][/tex]
[tex]\[ \frac{1}{6} \approx 0.16666666666666666 \][/tex]
When summed:
[tex]\[ \frac{1}{3126} + \frac{1}{6} \approx 0.16698656429942418 \][/tex]
