To find the mean of [tex]\(10, y^{-1}, 12, x, z^5\)[/tex] given that the mean of [tex]\(x, y^{-1}, z^5\)[/tex] is 6, let's follow the steps methodically.
### Step 1: Understand the Given Mean
We are given that the mean of [tex]\(x, y^{-1}, z^5\)[/tex] is 6. Mathematically, this can be expressed as:
[tex]\[
\frac{x + y^{-1} + z^5}{3} = 6
\][/tex]
### Step 2: Calculate the Combined Sum
To get the combined sum of [tex]\(x, y^{-1}, z^5\)[/tex], multiply both sides of the equation by 3:
[tex]\[
x + y^{-1} + z^5 = 6 \times 3 = 18
\][/tex]
So, the sum of [tex]\(x, y^{-1}, z^5\)[/tex] is 18.
### Step 3: Form the New Set and Calculate the Sum
Now, let's add the additional elements, 10 and 12, to form our new set: [tex]\(10, y^{-1}, 12, x, z^5\)[/tex].
The sum of this new set can be calculated by adding the fixed elements 10 and 12 to our combined sum from Step 2:
[tex]\[
10 + 12 + x + y^{-1} + z^5 = 10 + 12 + 18
\][/tex]
[tex]\[
10 + 12 + 18 = 40
\][/tex]
### Step 4: Calculate the Mean of the New Set
The new set contains 5 elements. To find the mean, divide the total sum by the number of elements in the set:
[tex]\[
\text{Mean} = \frac{10 + y^{-1} + 12 + x + z^5}{5} = \frac{40}{5} = 8
\][/tex]
### Conclusion
The mean of the set [tex]\(10, y^{-1}, 12, x, z^5\)[/tex] is [tex]\(\boxed{8}\)[/tex].
