Simplify the Radical Expression 10 square root of 20-12 square root of 45+9 square root of 24-15 square root of 54

$10 \sqrt{20} - 12 \sqrt{45} + 9 \sqrt{24} - 15 \sqrt{54}$

Step 1

Simplify each term.

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

Factor $4$ out of $20$ .

$10 \sqrt{4 (5)} - 12 \sqrt{45} + 9 \sqrt{24} - 15 \sqrt{54}$

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

$10 \sqrt{2^{2} \cdot 5} - 12 \sqrt{45} + 9 \sqrt{24} - 15 \sqrt{54}$

Pull terms out from under the radical.

$10 (| 2 | \sqrt{5}) - 12 \sqrt{45} + 9 \sqrt{24} - 15 \sqrt{54}$

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$10 (2 \sqrt{5}) - 12 \sqrt{45} + 9 \sqrt{24} - 15 \sqrt{54}$

Multiply $2$ by $10$ .

$20 \sqrt{5} - 12 \sqrt{45} + 9 \sqrt{24} - 15 \sqrt{54}$

Rewrite $45$ as $3^{2} \cdot 5$ .

Factor $9$ out of $45$ .

$20 \sqrt{5} - 12 \sqrt{9 (5)} + 9 \sqrt{24} - 15 \sqrt{54}$

Rewrite $9$ as $3^{2}$ .

$20 \sqrt{5} - 12 \sqrt{3^{2} \cdot 5} + 9 \sqrt{24} - 15 \sqrt{54}$

Pull terms out from under the radical.

$20 \sqrt{5} - 12 (| 3 | \sqrt{5}) + 9 \sqrt{24} - 15 \sqrt{54}$

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$20 \sqrt{5} - 12 (3 \sqrt{5}) + 9 \sqrt{24} - 15 \sqrt{54}$

Multiply $3$ by $- 12$ .

$20 \sqrt{5} - 36 \sqrt{5} + 9 \sqrt{24} - 15 \sqrt{54}$

Rewrite $24$ as $2^{2} \cdot 6$ .

Factor $4$ out of $24$ .

$20 \sqrt{5} - 36 \sqrt{5} + 9 \sqrt{4 (6)} - 15 \sqrt{54}$

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

$20 \sqrt{5} - 36 \sqrt{5} + 9 \sqrt{2^{2} \cdot 6} - 15 \sqrt{54}$

Pull terms out from under the radical.

$20 \sqrt{5} - 36 \sqrt{5} + 9 (| 2 | \sqrt{6}) - 15 \sqrt{54}$

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$20 \sqrt{5} - 36 \sqrt{5} + 9 (2 \sqrt{6}) - 15 \sqrt{54}$

Multiply $2$ by $9$ .

$20 \sqrt{5} - 36 \sqrt{5} + 18 \sqrt{6} - 15 \sqrt{54}$

Rewrite $54$ as $3^{2} \cdot 6$ .

Factor $9$ out of $54$ .

$20 \sqrt{5} - 36 \sqrt{5} + 18 \sqrt{6} - 15 \sqrt{9 (6)}$

Rewrite $9$ as $3^{2}$ .

$20 \sqrt{5} - 36 \sqrt{5} + 18 \sqrt{6} - 15 \sqrt{3^{2} \cdot 6}$

Pull terms out from under the radical.

$20 \sqrt{5} - 36 \sqrt{5} + 18 \sqrt{6} - 15 (| 3 | \sqrt{6})$

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$20 \sqrt{5} - 36 \sqrt{5} + 18 \sqrt{6} - 15 (3 \sqrt{6})$

Multiply $3$ by $- 15$ .

$20 \sqrt{5} - 36 \sqrt{5} + 18 \sqrt{6} - 45 \sqrt{6}$

Step 2

Subtract $36 \sqrt{5}$ from $20 \sqrt{5}$ .

$- 16 \sqrt{5} + 18 \sqrt{6} - 45 \sqrt{6}$

Step 3

Subtract $45 \sqrt{6}$ from $18 \sqrt{6}$ .

$- 16 \sqrt{5} - 27 \sqrt{6}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$- 16 \sqrt{5} - 27 \sqrt{6}$

Decimal Form:

$- 101.91331069 \dots$

Simplify the Radical Expression 10 square root of 20-12 square root of 45+9 square root of 24-15 square root of 54

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