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AP DSC School Assistant Math Model Test 1

Section: Subject Specific

Question : 85 of 160

Marks: +1, -0

What is the geometric mean of 6, 8, 16 and 27?

Solution:

Concept:

The geometric mean of a set of n numbers is the n-th root of their product. For numbers

a_1, a_2, \dots, a_n

, it is given by

\text{GM} = (a_1 \times a_2 \times \cdots \times a_n)^{1/n}

Explanation:

Here

n = 4

and the numbers are 6, 8, 16, and 27.

First, find the product:

6 \times 8 \times 16 \times 27

Factor each number into primes:

6 = 2 \times 3

8 = 2^3

16 = 2^4

27 = 3^3

Product

= 2^{1+3+4} \times 3^{1+3} = 2^8 \times 3^4

Now, take the 4th root:

\text{GM} = (2^8 \times 3^4)^{1/4} = 2^{8/4} \times 3^{4/4} = 2^2 \times 3^1 = 4 \times 3 = 12

Answer:

The geometric mean is 12, which corresponds to option B.

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