What is the geometric mean of 7 and 12?
Understand the Problem
The question is asking for the geometric mean of the numbers 7 and 12. The geometric mean is calculated by taking the square root of the product of the two numbers. In this case, we will multiply 7 and 12 together and then take the square root of the result.
Answer
The geometric mean of 7 and 12 is \( 2\sqrt{21} \).
Answer for screen readers
The geometric mean of 7 and 12 is ( 2\sqrt{21} ).
Steps to Solve
- Multiply the Numbers
To find the geometric mean, we start by multiplying the two numbers together.
$$ 7 \times 12 = 84 $$
- Take the Square Root
Next, we take the square root of the product we found in the first step.
$$ \sqrt{84} $$
- Simplify the Square Root
Now we simplify the square root. The number 84 can be factored.
Since ( 84 = 4 \times 21 ), we can simplify:
$$ \sqrt{84} = \sqrt{4 \times 21} = \sqrt{4} \times \sqrt{21} = 2\sqrt{21} $$
- Final Value
The geometric mean is the simplified result, which is ( 2\sqrt{21} ).
The geometric mean of 7 and 12 is ( 2\sqrt{21} ).
More Information
The geometric mean is particularly useful in various fields such as finance and biology, especially when comparing different items with different properties. It is more meaningful than the arithmetic mean when dealing with ratios or percentages.
Tips
- Forgetting to take the square root after multiplying the numbers.
- Not simplifying the square root properly. Always check for perfect squares that can be factored out of the square root.
