Basic Math Examples

$2 \sqrt{50}$ , $4 \sqrt{72}$ , $6 \sqrt{98}$ , $2 \sqrt{20}$

Step 1

Rewrite $50$ as $5^{2} \cdot 2$ .

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Step 1.1

Factor $25$ out of $50$ .

$2 \sqrt{25 (2)}, 4 \sqrt{72}, 6 \sqrt{98}, 2 \sqrt{20}$

Step 1.2

Rewrite $25$ as $5^{2}$ .

$2 \sqrt{5^{2} \cdot 2}, 4 \sqrt{72}, 6 \sqrt{98}, 2 \sqrt{20}$

Step 2

Pull terms out from under the radical.

$2 (5 \sqrt{2}), 4 \sqrt{72}, 6 \sqrt{98}, 2 \sqrt{20}$

Step 3

Multiply $5$ by $2$ .

$10 \sqrt{2}, 4 \sqrt{72}, 6 \sqrt{98}, 2 \sqrt{20}$

Step 4

Rewrite $72$ as $6^{2} \cdot 2$ .

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Step 4.1

Factor $36$ out of $72$ .

$10 \sqrt{2}, 4 \sqrt{36 (2)}, 6 \sqrt{98}, 2 \sqrt{20}$

Step 4.2

Rewrite $36$ as $6^{2}$ .

$10 \sqrt{2}, 4 \sqrt{6^{2} \cdot 2}, 6 \sqrt{98}, 2 \sqrt{20}$

Step 5

Pull terms out from under the radical.

$10 \sqrt{2}, 4 (6 \sqrt{2}), 6 \sqrt{98}, 2 \sqrt{20}$

Step 6

Multiply $6$ by $4$ .

$10 \sqrt{2}, 24 \sqrt{2}, 6 \sqrt{98}, 2 \sqrt{20}$

Step 7

Rewrite $98$ as $7^{2} \cdot 2$ .

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Step 7.1

Factor $49$ out of $98$ .

$10 \sqrt{2}, 24 \sqrt{2}, 6 \sqrt{49 (2)}, 2 \sqrt{20}$

Step 7.2

Rewrite $49$ as $7^{2}$ .

$10 \sqrt{2}, 24 \sqrt{2}, 6 \sqrt{7^{2} \cdot 2}, 2 \sqrt{20}$

Step 8

Pull terms out from under the radical.

$10 \sqrt{2}, 24 \sqrt{2}, 6 (7 \sqrt{2}), 2 \sqrt{20}$

Step 9

Multiply $7$ by $6$ .

$10 \sqrt{2}, 24 \sqrt{2}, 42 \sqrt{2}, 2 \sqrt{20}$

Step 10

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 10.1

Factor $4$ out of $20$ .

$10 \sqrt{2}, 24 \sqrt{2}, 42 \sqrt{2}, 2 \sqrt{4 (5)}$

Step 10.2

Rewrite $4$ as $2^{2}$ .

$10 \sqrt{2}, 24 \sqrt{2}, 42 \sqrt{2}, 2 \sqrt{2^{2} \cdot 5}$

Step 11

Pull terms out from under the radical.

$10 \sqrt{2}, 24 \sqrt{2}, 42 \sqrt{2}, 2 (2 \sqrt{5})$

Step 12

Multiply $2$ by $2$ .

$10 \sqrt{2}, 24 \sqrt{2}, 42 \sqrt{2}, 4 \sqrt{5}$

Step 13

Use the formula to find the geometric mean.

$\sqrt[4]{10 \sqrt{2} \cdot (24 \sqrt{2}) \cdot (42 \sqrt{2}) \cdot (4 \sqrt{5})}$

Step 14

Multiply $24$ by $10$ .

$\sqrt[4]{240 \sqrt{2} \sqrt{2} \cdot 42 \sqrt{2} \cdot 4 \sqrt{5}}$

Step 15

Multiply $42$ by $240$ .

$\sqrt[4]{10080 \sqrt{2} \sqrt{2} \sqrt{2} \cdot 4 \sqrt{5}}$

Step 16

Multiply $4$ by $10080$ .

$\sqrt[4]{40320 \sqrt{2} \sqrt{2} \sqrt{2} \sqrt{5}}$

Step 17

Raise $\sqrt{2}$ to the power of $1$ .

$\sqrt[4]{40320 ({\sqrt{2}}^{1} \sqrt{2}) \sqrt{2} \sqrt{5}}$

Step 18

Raise $\sqrt{2}$ to the power of $1$ .

$\sqrt[4]{40320 ({\sqrt{2}}^{1} {\sqrt{2}}^{1}) \sqrt{2} \sqrt{5}}$

Step 19

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sqrt[4]{40320 {\sqrt{2}}^{1 + 1} \sqrt{2} \sqrt{5}}$

Step 20

Add $1$ and $1$ .

$\sqrt[4]{40320 {\sqrt{2}}^{2} \sqrt{2} \sqrt{5}}$

Step 21

Raise $\sqrt{2}$ to the power of $1$ .

$\sqrt[4]{40320 ({\sqrt{2}}^{1} {\sqrt{2}}^{2}) \sqrt{5}}$

Step 22

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sqrt[4]{40320 {\sqrt{2}}^{1 + 2} \sqrt{5}}$

Step 23

Add $1$ and $2$ .

$\sqrt[4]{40320 {\sqrt{2}}^{3} \sqrt{5}}$

Step 24

Rewrite ${\sqrt{2}}^{3}$ as $\sqrt{2^{3}}$ .

$\sqrt[4]{40320 \sqrt{2^{3}} \sqrt{5}}$

Step 25

Raise $2$ to the power of $3$ .

$\sqrt[4]{40320 \sqrt{8} \sqrt{5}}$

Step 26

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 26.1

Factor $4$ out of $8$ .

$\sqrt[4]{40320 \sqrt{4 (2)} \sqrt{5}}$

Step 26.2

Rewrite $4$ as $2^{2}$ .

$\sqrt[4]{40320 \sqrt{2^{2} \cdot 2} \sqrt{5}}$

Step 27

Pull terms out from under the radical.

$\sqrt[4]{40320 \cdot 2 \sqrt{2} \sqrt{5}}$

Step 28

Combine exponents.

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Step 28.1

Multiply $40320$ by $2$ .

$\sqrt[4]{80640 \sqrt{2} \sqrt{5}}$

Step 28.2

Combine using the product rule for radicals.

$\sqrt[4]{80640 \sqrt{5 \cdot 2}}$

Step 28.3

Multiply $5$ by $2$ .

$\sqrt[4]{80640 \sqrt{10}}$

Step 29

Rewrite $80640 \sqrt{10}$ as $4^{4} \cdot (315 \sqrt{10})$ .

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Step 29.1

Factor $256$ out of $80640$ .

$\sqrt[4]{256 (315) \sqrt{10}}$

Step 29.2

Rewrite $256$ as $4^{4}$ .

$\sqrt[4]{4^{4} \cdot 315 \sqrt{10}}$

Step 29.3

Add parentheses.

$\sqrt[4]{4^{4} \cdot (315 \sqrt{10})}$

Step 30

Pull terms out from under the radical.

$4 \sqrt[4]{315 \sqrt{10}}$

Step 31

Approximate the result.

$22.47178803$

Step 32

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.

$22.5$