Simplify ( square root of 48)/( square root of 14)

$\frac{\sqrt{48}}{\sqrt{14}}$

Step 1

Combine $\sqrt{48}$ and $\sqrt{14}$ into a single radical.

$\sqrt{\frac{48}{14}}$

Step 2

Cancel the common factor of $48$ and $14$ .

Factor $2$ out of $48$ .

$\sqrt{\frac{2 (24)}{14}}$

Cancel the common factors.

Factor $2$ out of $14$ .

$\sqrt{\frac{2 \cdot 24}{2 \cdot 7}}$

Cancel the common factor.

$\sqrt{\frac{2 \cdot 24}{2 \cdot 7}}$

Rewrite the expression.

$\sqrt{\frac{24}{7}}$

Step 3

Rewrite $\sqrt{\frac{24}{7}}$ as $\frac{\sqrt{24}}{\sqrt{7}}$ .

$\frac{\sqrt{24}}{\sqrt{7}}$

Step 4

Simplify the numerator.

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

Factor $4$ out of $24$ .

$\frac{\sqrt{4 (6)}}{\sqrt{7}}$

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

$\frac{\sqrt{2^{2} \cdot 6}}{\sqrt{7}}$

Pull terms out from under the radical.

$\frac{2 \sqrt{6}}{\sqrt{7}}$

Step 5

Multiply $\frac{2 \sqrt{6}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$\frac{2 \sqrt{6}}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}}$

Step 6

Combine and simplify the denominator.

Multiply $\frac{2 \sqrt{6}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$\frac{2 \sqrt{6} \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Raise $\sqrt{7}$ to the power of $1$ .

$\frac{2 \sqrt{6} \sqrt{7}}{{\sqrt{7}}^{1} \sqrt{7}}$

Raise $\sqrt{7}$ to the power of $1$ .

$\frac{2 \sqrt{6} \sqrt{7}}{{\sqrt{7}}^{1} {\sqrt{7}}^{1}}$

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

$\frac{2 \sqrt{6} \sqrt{7}}{{\sqrt{7}}^{1 + 1}}$

Add $1$ and $1$ .

$\frac{2 \sqrt{6} \sqrt{7}}{{\sqrt{7}}^{2}}$

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$\frac{2 \sqrt{6} \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}}$

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

$\frac{2 \sqrt{6} \sqrt{7}}{7^{\frac{1}{2} \cdot 2}}$

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

$\frac{2 \sqrt{6} \sqrt{7}}{7^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 \sqrt{6} \sqrt{7}}{7^{\frac{2}{2}}}$

Rewrite the expression.

$\frac{2 \sqrt{6} \sqrt{7}}{7^{1}}$

Evaluate the exponent.

$\frac{2 \sqrt{6} \sqrt{7}}{7}$

Step 7

Simplify the numerator.

Combine using the product rule for radicals.

$\frac{2 \sqrt{7 \cdot 6}}{7}$

Multiply $7$ by $6$ .

$\frac{2 \sqrt{42}}{7}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{2 \sqrt{42}}{7}$

Decimal Form:

$1.85164019 \dots$

Simplify ( square root of 48)/( square root of 14)

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