Find the Geometric Mean 10 , 16 , 15 , 14 , 8 , 21 , 10 , 5 , 19 , 18 , 4 , 5 , 16 , 12 , 10 , 9

$10$ , $16$ , $15$ , $14$ , $8$ , $21$ , $10$ , $5$ , $19$ , $18$ , $4$ , $5$ , $16$ , $12$ , $10$ , $9$

Step 1

Use the formula to find the geometric mean.

$\sqrt[16]{10 \cdot 16 \cdot 15 \cdot 14 \cdot 8 \cdot 21 \cdot 10 \cdot 5 \cdot 19 \cdot 18 \cdot 4 \cdot 5 \cdot 16 \cdot 12 \cdot 10 \cdot 9}$

Step 2

Multiply $10$ by $16$ .

$\sqrt[16]{160 \cdot 15 \cdot 14 \cdot 8 \cdot 21 \cdot 10 \cdot 5 \cdot 19 \cdot 18 \cdot 4 \cdot 5 \cdot 16 \cdot 12 \cdot 10 \cdot 9}$

Step 3

Multiply $160$ by $15$ .

$\sqrt[16]{2400 \cdot 14 \cdot 8 \cdot 21 \cdot 10 \cdot 5 \cdot 19 \cdot 18 \cdot 4 \cdot 5 \cdot 16 \cdot 12 \cdot 10 \cdot 9}$

Step 4

Multiply $2400$ by $14$ .

$\sqrt[16]{33600 \cdot 8 \cdot 21 \cdot 10 \cdot 5 \cdot 19 \cdot 18 \cdot 4 \cdot 5 \cdot 16 \cdot 12 \cdot 10 \cdot 9}$

Step 5

Multiply $33600$ by $8$ .

$\sqrt[16]{268800 \cdot 21 \cdot 10 \cdot 5 \cdot 19 \cdot 18 \cdot 4 \cdot 5 \cdot 16 \cdot 12 \cdot 10 \cdot 9}$

Step 6

Multiply $268800$ by $21$ .

$\sqrt[16]{5644800 \cdot 10 \cdot 5 \cdot 19 \cdot 18 \cdot 4 \cdot 5 \cdot 16 \cdot 12 \cdot 10 \cdot 9}$

Step 7

Multiply $5644800$ by $10$ .

$\sqrt[16]{56448000 \cdot 5 \cdot 19 \cdot 18 \cdot 4 \cdot 5 \cdot 16 \cdot 12 \cdot 10 \cdot 9}$

Step 8

Multiply $56448000$ by $5$ .

$\sqrt[16]{282240000 \cdot 19 \cdot 18 \cdot 4 \cdot 5 \cdot 16 \cdot 12 \cdot 10 \cdot 9}$

Step 9

Multiply $282240000$ by $19$ .

$\sqrt[16]{5362560000 \cdot 18 \cdot 4 \cdot 5 \cdot 16 \cdot 12 \cdot 10 \cdot 9}$

Step 10

Multiply $5362560000$ by $18$ .

$\sqrt[16]{96526080000 \cdot 4 \cdot 5 \cdot 16 \cdot 12 \cdot 10 \cdot 9}$

Step 11

Multiply $96526080000$ by $4$ .

$\sqrt[16]{386104320000 \cdot 5 \cdot 16 \cdot 12 \cdot 10 \cdot 9}$

Step 12

Multiply $386104320000$ by $5$ .

$\sqrt[16]{1930521600000 \cdot 16 \cdot 12 \cdot 10 \cdot 9}$

Step 13

Multiply $1930521600000$ by $16$ .

$\sqrt[16]{30888345600000 \cdot 12 \cdot 10 \cdot 9}$

Step 14

Multiply $30888345600000$ by $12$ .

$\sqrt[16]{370660147200000 \cdot 10 \cdot 9}$

Step 15

Multiply $370660147200000$ by $10$ .

$\sqrt[16]{3706601472000000 \cdot 9}$

Step 16

Multiply $3706601472000000$ by $9$ .

$\sqrt[16]{33359413248000000}$

Step 17

Rewrite $33359413248000000$ as $2^{16} \cdot 509024250000$ .

Factor $65536$ out of $33359413248000000$ .

$\sqrt[16]{65536 (509024250000)}$

Rewrite $65536$ as $2^{16}$ .

$\sqrt[16]{2^{16} \cdot 509024250000}$

Step 18

Pull terms out from under the radical.

$2 \sqrt[16]{509024250000}$

Step 19

Approximate the result.

$10.78204525$

Step 20

The geometric mean should be rounded to one more decimal place than the original data. If the original data were mixed, round to one decimal place more than the least precise.

$10.8$

Find the Geometric Mean 10 , 16 , 15 , 14 , 8 , 21 , 10 , 5 , 19 , 18 , 4 , 5 , 16 , 12 , 10 , 9

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