To find the geometric mean of two numbers, you need to take the square root of their product. Mathematically, the geometric mean [tex]\( G \)[/tex] of numbers [tex]\( a \)[/tex] and [tex]\( b \)[/tex] is given by:
[tex]\[ G = \sqrt{a \cdot b} \][/tex]
Given the numbers 7 and 9, we can calculate their geometric mean as follows:
1. Calculate the product of 7 and 9:
[tex]\[ 7 \times 9 = 63 \][/tex]
2. Take the square root of 63 to find the geometric mean:
[tex]\[ \sqrt{63} \][/tex]
We need to determine which of the given choices matches [tex]\( \sqrt{63} \)[/tex].
Let's evaluate each option:
A. [tex]\( 3 \sqrt{7} \)[/tex]
[tex]\[ 3 \sqrt{7} \approx 3 \times 2.6458 = 7.9374 \][/tex]
B. [tex]\( \sqrt{70} \)[/tex]
[tex]\[ \sqrt{70} \approx 8.3666 \][/tex]
C. [tex]\( 6 \sqrt{2} \)[/tex]
[tex]\[ 6 \sqrt{2} \approx 6 \times 1.4142 = 8.4852 \][/tex]
D. 63
This is clearly not [tex]\( \sqrt{63} \)[/tex].
To confirm the correct answer, let us approximate [tex]\( \sqrt{63} \)[/tex]:
The value of [tex]\( \sqrt{63} \)[/tex] is approximately 7.9373.
From the approximations, [tex]\( 3 \sqrt{7} \)[/tex] approximately equals 7.9374, which is very close to our calculated geometric mean of 7.9373 (considering floating-point tolerance).
Therefore, the best choice from the given options that represents [tex]\( \sqrt{63} \)[/tex] is:
A. [tex]\( 3 \sqrt{7} \)[/tex]
