Pre-Algebra Examples

$3 \sqrt{125}$ , $2 \sqrt{180}$ , $\sqrt{147}$

Step 1

Rewrite $125$ as $5^{2} \cdot 5$ .

Tap for more steps...

Step 1.1

Factor $25$ out of $125$ .

$3 \sqrt{25 (5)}, 2 \sqrt{180}, \sqrt{147}$

Step 1.2

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

$3 \sqrt{5^{2} \cdot 5}, 2 \sqrt{180}, \sqrt{147}$

Step 2

Pull terms out from under the radical.

$3 (5 \sqrt{5}), 2 \sqrt{180}, \sqrt{147}$

Step 3

Multiply $5$ by $3$ .

$15 \sqrt{5}, 2 \sqrt{180}, \sqrt{147}$

Step 4

Rewrite $180$ as $6^{2} \cdot 5$ .

Tap for more steps...

Step 4.1

Factor $36$ out of $180$ .

$15 \sqrt{5}, 2 \sqrt{36 (5)}, \sqrt{147}$

Step 4.2

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

$15 \sqrt{5}, 2 \sqrt{6^{2} \cdot 5}, \sqrt{147}$

Step 5

Pull terms out from under the radical.

$15 \sqrt{5}, 2 (6 \sqrt{5}), \sqrt{147}$

Step 6

Multiply $6$ by $2$ .

$15 \sqrt{5}, 12 \sqrt{5}, \sqrt{147}$

Step 7

Rewrite $147$ as $7^{2} \cdot 3$ .

Tap for more steps...

Step 7.1

Factor $49$ out of $147$ .

$15 \sqrt{5}, 12 \sqrt{5}, \sqrt{49 (3)}$

Step 7.2

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

$15 \sqrt{5}, 12 \sqrt{5}, \sqrt{7^{2} \cdot 3}$

Step 8

Pull terms out from under the radical.

$15 \sqrt{5}, 12 \sqrt{5}, 7 \sqrt{3}$

Step 9

Use the formula to find the geometric mean.

$\sqrt[3]{15 \sqrt{5} \cdot (12 \sqrt{5}) \cdot (7 \sqrt{3})}$

Step 10

Multiply $12$ by $15$ .

$\sqrt[3]{180 \sqrt{5} \sqrt{5} \cdot 7 \sqrt{3}}$

Step 11

Multiply $7$ by $180$ .

$\sqrt[3]{1260 \sqrt{5} \sqrt{5} \sqrt{3}}$

Step 12

Raise $\sqrt{5}$ to the power of $1$ .

$\sqrt[3]{1260 ({\sqrt{5}}^{1} \sqrt{5}) \sqrt{3}}$

Step 13

Raise $\sqrt{5}$ to the power of $1$ .

$\sqrt[3]{1260 ({\sqrt{5}}^{1} {\sqrt{5}}^{1}) \sqrt{3}}$

Step 14

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

$\sqrt[3]{1260 {\sqrt{5}}^{1 + 1} \sqrt{3}}$

Step 15

Add $1$ and $1$ .

$\sqrt[3]{1260 {\sqrt{5}}^{2} \sqrt{3}}$

Step 16

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

Tap for more steps...

Step 16.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\sqrt[3]{1260 {(5^{\frac{1}{2}})}^{2} \sqrt{3}}$

Step 16.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sqrt[3]{1260 \cdot 5^{\frac{1}{2} \cdot 2} \sqrt{3}}$

Step 16.3

Combine $\frac{1}{2}$ and $2$ .

$\sqrt[3]{1260 \cdot 5^{\frac{2}{2}} \sqrt{3}}$

Step 16.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 16.4.1

Cancel the common factor.

$\sqrt[3]{1260 \cdot 5^{\frac{2}{2}} \sqrt{3}}$

Step 16.4.2

Rewrite the expression.

$\sqrt[3]{1260 \cdot 5^{1} \sqrt{3}}$

Step 16.5

Evaluate the exponent.

$\sqrt[3]{1260 \cdot 5 \sqrt{3}}$

Step 17

Multiply $1260$ by $5$ .

$\sqrt[3]{6300 \sqrt{3}}$

Step 18

Approximate the result.

$22.18028176$

Step 19

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.

$22.2$