Simplify ((5 square root of 96)/6)÷((2 square root of 24)/5)

$\frac{5 \sqrt{96}}{6} \div \frac{2 \sqrt{24}}{5}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{5 \sqrt{96}}{6} \cdot \frac{5}{2 \sqrt{24}}$

Step 2

Simplify the numerator.

Rewrite $96$ as $4^{2} \cdot 6$ .

Factor $16$ out of $96$ .

$\frac{5 \sqrt{16 (6)}}{6} \cdot \frac{5}{2 \sqrt{24}}$

Rewrite $16$ as $4^{2}$ .

$\frac{5 \sqrt{4^{2} \cdot 6}}{6} \cdot \frac{5}{2 \sqrt{24}}$

Pull terms out from under the radical.

$\frac{5 \cdot 4 \sqrt{6}}{6} \cdot \frac{5}{2 \sqrt{24}}$

Multiply $5$ by $4$ .

$\frac{20 \sqrt{6}}{6} \cdot \frac{5}{2 \sqrt{24}}$

Step 3

Simplify terms.

Cancel the common factor of $2$ .

Factor $2$ out of $20 \sqrt{6}$ .

$\frac{2 (10 \sqrt{6})}{6} \cdot \frac{5}{2 \sqrt{24}}$

Factor $2$ out of $2 \sqrt{24}$ .

$\frac{2 (10 \sqrt{6})}{6} \cdot \frac{5}{2 (\sqrt{24})}$

Cancel the common factor.

$\frac{2 (10 \sqrt{6})}{6} \cdot \frac{5}{2 \sqrt{24}}$

Rewrite the expression.

$\frac{10 \sqrt{6}}{6} \cdot \frac{5}{\sqrt{24}}$

Multiply $\frac{10 \sqrt{6}}{6}$ by $\frac{5}{\sqrt{24}}$ .

$\frac{10 \sqrt{6} \cdot 5}{6 \sqrt{24}}$

Multiply $5$ by $10$ .

$\frac{50 \sqrt{6}}{6 \sqrt{24}}$

Cancel the common factor of $50$ and $6$ .

Factor $2$ out of $50 \sqrt{6}$ .

$\frac{2 (25 \sqrt{6})}{6 \sqrt{24}}$

Cancel the common factors.

Factor $2$ out of $6 \sqrt{24}$ .

$\frac{2 (25 \sqrt{6})}{2 (3 \sqrt{24})}$

Cancel the common factor.

$\frac{2 (25 \sqrt{6})}{2 (3 \sqrt{24})}$

Rewrite the expression.

$\frac{25 \sqrt{6}}{3 \sqrt{24}}$

Step 4

Combine $\sqrt{6}$ and $\sqrt{24}$ into a single radical.

$\frac{25 \sqrt{\frac{6}{24}}}{3}$

Step 5

Cancel the common factor of $6$ and $24$ .

Factor $6$ out of $6$ .

$\frac{25 \sqrt{\frac{6 (1)}{24}}}{3}$

Cancel the common factors.

Factor $6$ out of $24$ .

$\frac{25 \sqrt{\frac{6 \cdot 1}{6 \cdot 4}}}{3}$

Cancel the common factor.

$\frac{25 \sqrt{\frac{6 \cdot 1}{6 \cdot 4}}}{3}$

Rewrite the expression.

$\frac{25 \sqrt{\frac{1}{4}}}{3}$

Step 6

Simplify the numerator.

Rewrite $\sqrt{\frac{1}{4}}$ as $\frac{\sqrt{1}}{\sqrt{4}}$ .

$\frac{25 \frac{\sqrt{1}}{\sqrt{4}}}{3}$

Any root of $1$ is $1$ .

$\frac{25 \frac{1}{\sqrt{4}}}{3}$

Simplify the denominator.

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

$\frac{25 \frac{1}{\sqrt{2^{2}}}}{3}$

Pull terms out from under the radical, assuming positive real numbers.

$\frac{25 (\frac{1}{2})}{3}$

Step 7

Combine $25$ and $\frac{1}{2}$ .

$\frac{\frac{25}{2}}{3}$

Step 8

Multiply the numerator by the reciprocal of the denominator.

$\frac{25}{2} \cdot \frac{1}{3}$

Step 9

Multiply $\frac{25}{2} \cdot \frac{1}{3}$ .

Multiply $\frac{25}{2}$ by $\frac{1}{3}$ .

$\frac{25}{2 \cdot 3}$

Multiply $2$ by $3$ .

$\frac{25}{6}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{25}{6}$

Decimal Form:

$4.1\overline{6}$

Mixed Number Form:

$4 \frac{1}{6}$

Simplify ((5 square root of 96)/6)÷((2 square root of 24)/5)

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