To find the geometric mean of the numbers 5, 8, 10, 15, 20, 25, 30, and 35, we can follow these steps:
### Step 1: Understand the formula for the geometric mean
The geometric mean (GM) of a set of n numbers \( x_1, x_2, ..., x_n \) is given by the formula:
\[
GM = (x_1 \times x_2 \times ... \times x_n)^{\frac{1}{n}}
\]
### Step 2: Identify the numbers
The given numbers are:
5, 8, 10, 15, 20, 25, 30, 35
### Step 3: Count the number of values
There are 8 values in total:
- \( n = 8 \)
### Step 4: Calculate the product of the numbers
Now we calculate the product of these numbers:
\[
5 \times 8 \times 10 \times 15 \times 20 \times 25 \times 30 \times 35
\]
### Step 5: Factor the numbers into their prime factors
To simplify the multiplication, we can express each number in terms of its prime factors:
- \( 5 = 5^1 \)
- \( 8 = 2^3 \)
- \( 10 = 2^1 \times 5^1 \)
- \( 15 = 3^1 \times 5^1 \)
- \( 20 = 2^2 \times 5^1 \)
- \( 25 = 5^2 \)
- \( 30 = 2^1 \times 3^1 \times 5^1 \)
- \( 35 = 5^1 \times 7^1 \)
### Step 6: Combine the prime factors
Now we combine the prime factors:
- Count the powers of each prime factor:
- For \( 2 \): \( 3 + 1 + 2 + 1 = 7 \) (from 8, 10, 20, 30)
- For \( 3 \): \( 1 + 1 = 2 \) (from 15, 30)
- For \( 5 \): \( 1 + 1 + 1 + 2 + 1 + 1 = 7 \) (from 5, 10, 15, 20, 25, 30, 35)
- For \( 7 \): \( 1 \) (from 35)
Thus, the product can be expressed as:
\[
2^7 \times 3^2 \times 5^7 \times 7^1
\]
### Step 7: Calculate the total product
Now we can write the product:
\[
P = 2^7 \times 3^2 \times 5^7 \times 7^1
\]
### Step 8: Apply the geometric mean formula
Now we apply the geometric mean formula:
\[
GM = (P)^{\frac{1}{8}} = (2^7 \times 3^2 \times 5^7 \times 7^1)^{\frac{1}{8}}
\]
### Step 9: Simplify the expression
Using the property of exponents:
\[
GM = 2^{\frac{7}{8}} \times 3^{\frac{2}{8}} \times 5^{\frac{7}{8}} \times 7^{\frac{1}{8}}
\]
### Step 10: Final Calculation
Now we can calculate the numerical value:
\[
GM \approx 10 \times (2^{\frac{7}{8}} \times 3^{\frac{1}{4}} \times 7^{\frac{1}{8}})
\]
### Conclusion
The final geometric mean can be approximated numerically or left in the exponent form as shown.
