Evaluate ( square root of 216- square root of 7)/( square root of 42-6)

$\frac{\sqrt{216} - \sqrt{7}}{\sqrt{42} - 6}$

Step 1

Simplify the numerator.

Rewrite $216$ as $6^{2} \cdot 6$ .

Factor $36$ out of $216$ .

$\frac{\sqrt{36 (6)} - \sqrt{7}}{\sqrt{42} - 6}$

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

$\frac{\sqrt{6^{2} \cdot 6} - \sqrt{7}}{\sqrt{42} - 6}$

Pull terms out from under the radical.

$\frac{6 \sqrt{6} - \sqrt{7}}{\sqrt{42} - 6}$

Step 2

Multiply $\frac{6 \sqrt{6} - \sqrt{7}}{\sqrt{42} - 6}$ by $\frac{\sqrt{42} + 6}{\sqrt{42} + 6}$ .

$\frac{6 \sqrt{6} - \sqrt{7}}{\sqrt{42} - 6} \cdot \frac{\sqrt{42} + 6}{\sqrt{42} + 6}$

Step 3

Combine fractions.

Multiply $\frac{6 \sqrt{6} - \sqrt{7}}{\sqrt{42} - 6}$ by $\frac{\sqrt{42} + 6}{\sqrt{42} + 6}$ .

$\frac{(6 \sqrt{6} - \sqrt{7}) (\sqrt{42} + 6)}{(\sqrt{42} - 6) (\sqrt{42} + 6)}$

Expand the denominator using the FOIL method.

$\frac{(6 \sqrt{6} - \sqrt{7}) (\sqrt{42} + 6)}{{\sqrt{42}}^{2} + \sqrt{42} \cdot 6 - 6 \sqrt{42} - 36}$

Simplify.

$\frac{(6 \sqrt{6} - \sqrt{7}) (\sqrt{42} + 6)}{6}$

Step 4

Expand $(6 \sqrt{6} - \sqrt{7}) (\sqrt{42} + 6)$ using the FOIL Method.

Apply the distributive property.

$\frac{6 \sqrt{6} (\sqrt{42} + 6) - \sqrt{7} (\sqrt{42} + 6)}{6}$

Apply the distributive property.

$\frac{6 \sqrt{6} \sqrt{42} + 6 \sqrt{6} \cdot 6 - \sqrt{7} (\sqrt{42} + 6)}{6}$

Apply the distributive property.

$\frac{6 \sqrt{6} \sqrt{42} + 6 \sqrt{6} \cdot 6 - \sqrt{7} \sqrt{42} - \sqrt{7} \cdot 6}{6}$

Step 5

Simplify and combine like terms.

Simplify each term.

Multiply $6 \sqrt{6} \sqrt{42}$ .

Combine using the product rule for radicals.

$\frac{6 \sqrt{42 \cdot 6} + 6 \sqrt{6} \cdot 6 - \sqrt{7} \sqrt{42} - \sqrt{7} \cdot 6}{6}$

Multiply $42$ by $6$ .

$\frac{6 \sqrt{252} + 6 \sqrt{6} \cdot 6 - \sqrt{7} \sqrt{42} - \sqrt{7} \cdot 6}{6}$

Rewrite $252$ as $6^{2} \cdot 7$ .

Factor $36$ out of $252$ .

$\frac{6 \sqrt{36 (7)} + 6 \sqrt{6} \cdot 6 - \sqrt{7} \sqrt{42} - \sqrt{7} \cdot 6}{6}$

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

$\frac{6 \sqrt{6^{2} \cdot 7} + 6 \sqrt{6} \cdot 6 - \sqrt{7} \sqrt{42} - \sqrt{7} \cdot 6}{6}$

Pull terms out from under the radical.

$\frac{6 (6 \sqrt{7}) + 6 \sqrt{6} \cdot 6 - \sqrt{7} \sqrt{42} - \sqrt{7} \cdot 6}{6}$

Multiply $6$ by $6$ .

$\frac{36 \sqrt{7} + 6 \sqrt{6} \cdot 6 - \sqrt{7} \sqrt{42} - \sqrt{7} \cdot 6}{6}$

Multiply $6$ by $6$ .

$\frac{36 \sqrt{7} + 36 \sqrt{6} - \sqrt{7} \sqrt{42} - \sqrt{7} \cdot 6}{6}$

Multiply $- \sqrt{7} \sqrt{42}$ .

Combine using the product rule for radicals.

$\frac{36 \sqrt{7} + 36 \sqrt{6} - \sqrt{42 \cdot 7} - \sqrt{7} \cdot 6}{6}$

Multiply $42$ by $7$ .

$\frac{36 \sqrt{7} + 36 \sqrt{6} - \sqrt{294} - \sqrt{7} \cdot 6}{6}$

Rewrite $294$ as $7^{2} \cdot 6$ .

Factor $49$ out of $294$ .

$\frac{36 \sqrt{7} + 36 \sqrt{6} - \sqrt{49 (6)} - \sqrt{7} \cdot 6}{6}$

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

$\frac{36 \sqrt{7} + 36 \sqrt{6} - \sqrt{7^{2} \cdot 6} - \sqrt{7} \cdot 6}{6}$

Pull terms out from under the radical.

$\frac{36 \sqrt{7} + 36 \sqrt{6} - (7 \sqrt{6}) - \sqrt{7} \cdot 6}{6}$

Multiply $7$ by $- 1$ .

$\frac{36 \sqrt{7} + 36 \sqrt{6} - 7 \sqrt{6} - \sqrt{7} \cdot 6}{6}$

Multiply $6$ by $- 1$ .

$\frac{36 \sqrt{7} + 36 \sqrt{6} - 7 \sqrt{6} - 6 \sqrt{7}}{6}$

Subtract $6 \sqrt{7}$ from $36 \sqrt{7}$ .

$\frac{30 \sqrt{7} + 36 \sqrt{6} - 7 \sqrt{6}}{6}$

Subtract $7 \sqrt{6}$ from $36 \sqrt{6}$ .

$\frac{30 \sqrt{7} + 29 \sqrt{6}}{6}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{30 \sqrt{7} + 29 \sqrt{6}}{6}$

Decimal Form:

$25.06795697 \dots$

Evaluate ( square root of 216- square root of 7)/( square root of 42-6)

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