Simplify ( square root of 125- square root of 7)/( square root of 35-5)

$\frac{\sqrt{125} - \sqrt{7}}{\sqrt{35} - 5}$

Step 1

Simplify the numerator.

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

Factor $25$ out of $125$ .

$\frac{\sqrt{25 (5)} - \sqrt{7}}{\sqrt{35} - 5}$

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

$\frac{\sqrt{5^{2} \cdot 5} - \sqrt{7}}{\sqrt{35} - 5}$

Pull terms out from under the radical.

$\frac{5 \sqrt{5} - \sqrt{7}}{\sqrt{35} - 5}$

Step 2

Multiply $\frac{5 \sqrt{5} - \sqrt{7}}{\sqrt{35} - 5}$ by $\frac{\sqrt{35} + 5}{\sqrt{35} + 5}$ .

$\frac{5 \sqrt{5} - \sqrt{7}}{\sqrt{35} - 5} \cdot \frac{\sqrt{35} + 5}{\sqrt{35} + 5}$

Step 3

Combine fractions.

Multiply $\frac{5 \sqrt{5} - \sqrt{7}}{\sqrt{35} - 5}$ by $\frac{\sqrt{35} + 5}{\sqrt{35} + 5}$ .

$\frac{(5 \sqrt{5} - \sqrt{7}) (\sqrt{35} + 5)}{(\sqrt{35} - 5) (\sqrt{35} + 5)}$

Expand the denominator using the FOIL method.

$\frac{(5 \sqrt{5} - \sqrt{7}) (\sqrt{35} + 5)}{{\sqrt{35}}^{2} + \sqrt{35} \cdot 5 - 5 \sqrt{35} - 25}$

Simplify.

$\frac{(5 \sqrt{5} - \sqrt{7}) (\sqrt{35} + 5)}{10}$

Step 4

Expand $(5 \sqrt{5} - \sqrt{7}) (\sqrt{35} + 5)$ using the FOIL Method.

Apply the distributive property.

$\frac{5 \sqrt{5} (\sqrt{35} + 5) - \sqrt{7} (\sqrt{35} + 5)}{10}$

Apply the distributive property.

$\frac{5 \sqrt{5} \sqrt{35} + 5 \sqrt{5} \cdot 5 - \sqrt{7} (\sqrt{35} + 5)}{10}$

Apply the distributive property.

$\frac{5 \sqrt{5} \sqrt{35} + 5 \sqrt{5} \cdot 5 - \sqrt{7} \sqrt{35} - \sqrt{7} \cdot 5}{10}$

Step 5

Simplify and combine like terms.

Simplify each term.

Multiply $5 \sqrt{5} \sqrt{35}$ .

Combine using the product rule for radicals.

$\frac{5 \sqrt{35 \cdot 5} + 5 \sqrt{5} \cdot 5 - \sqrt{7} \sqrt{35} - \sqrt{7} \cdot 5}{10}$

Multiply $35$ by $5$ .

$\frac{5 \sqrt{175} + 5 \sqrt{5} \cdot 5 - \sqrt{7} \sqrt{35} - \sqrt{7} \cdot 5}{10}$

Rewrite $175$ as $5^{2} \cdot 7$ .

Factor $25$ out of $175$ .

$\frac{5 \sqrt{25 (7)} + 5 \sqrt{5} \cdot 5 - \sqrt{7} \sqrt{35} - \sqrt{7} \cdot 5}{10}$

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

$\frac{5 \sqrt{5^{2} \cdot 7} + 5 \sqrt{5} \cdot 5 - \sqrt{7} \sqrt{35} - \sqrt{7} \cdot 5}{10}$

Pull terms out from under the radical.

$\frac{5 (5 \sqrt{7}) + 5 \sqrt{5} \cdot 5 - \sqrt{7} \sqrt{35} - \sqrt{7} \cdot 5}{10}$

Multiply $5$ by $5$ .

$\frac{25 \sqrt{7} + 5 \sqrt{5} \cdot 5 - \sqrt{7} \sqrt{35} - \sqrt{7} \cdot 5}{10}$

Multiply $5$ by $5$ .

$\frac{25 \sqrt{7} + 25 \sqrt{5} - \sqrt{7} \sqrt{35} - \sqrt{7} \cdot 5}{10}$

Multiply $- \sqrt{7} \sqrt{35}$ .

Combine using the product rule for radicals.

$\frac{25 \sqrt{7} + 25 \sqrt{5} - \sqrt{35 \cdot 7} - \sqrt{7} \cdot 5}{10}$

Multiply $35$ by $7$ .

$\frac{25 \sqrt{7} + 25 \sqrt{5} - \sqrt{245} - \sqrt{7} \cdot 5}{10}$

Rewrite $245$ as $7^{2} \cdot 5$ .

Factor $49$ out of $245$ .

$\frac{25 \sqrt{7} + 25 \sqrt{5} - \sqrt{49 (5)} - \sqrt{7} \cdot 5}{10}$

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

$\frac{25 \sqrt{7} + 25 \sqrt{5} - \sqrt{7^{2} \cdot 5} - \sqrt{7} \cdot 5}{10}$

Pull terms out from under the radical.

$\frac{25 \sqrt{7} + 25 \sqrt{5} - (7 \sqrt{5}) - \sqrt{7} \cdot 5}{10}$

Multiply $7$ by $- 1$ .

$\frac{25 \sqrt{7} + 25 \sqrt{5} - 7 \sqrt{5} - \sqrt{7} \cdot 5}{10}$

Multiply $5$ by $- 1$ .

$\frac{25 \sqrt{7} + 25 \sqrt{5} - 7 \sqrt{5} - 5 \sqrt{7}}{10}$

Subtract $5 \sqrt{7}$ from $25 \sqrt{7}$ .

$\frac{20 \sqrt{7} + 25 \sqrt{5} - 7 \sqrt{5}}{10}$

Subtract $7 \sqrt{5}$ from $25 \sqrt{5}$ .

$\frac{20 \sqrt{7} + 18 \sqrt{5}}{10}$

Step 6

Cancel the common factor of $20 \sqrt{7} + 18 \sqrt{5}$ and $10$ .

Factor $2$ out of $20 \sqrt{7}$ .

$\frac{2 (10 \sqrt{7}) + 18 \sqrt{5}}{10}$

Factor $2$ out of $18 \sqrt{5}$ .

$\frac{2 (10 \sqrt{7}) + 2 (9 \sqrt{5})}{10}$

Factor $2$ out of $2 (10 \sqrt{7}) + 2 (9 \sqrt{5})$ .

$\frac{2 (10 \sqrt{7} + 9 \sqrt{5})}{10}$

Cancel the common factors.

Factor $2$ out of $10$ .

$\frac{2 (10 \sqrt{7} + 9 \sqrt{5})}{2 (5)}$

Cancel the common factor.

$\frac{2 (10 \sqrt{7} + 9 \sqrt{5})}{2 \cdot 5}$

Rewrite the expression.

$\frac{10 \sqrt{7} + 9 \sqrt{5}}{5}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{10 \sqrt{7} + 9 \sqrt{5}}{5}$

Decimal Form:

$9.31642498 \dots$

Simplify ( square root of 125- square root of 7)/( square root of 35-5)

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