Simplify the Radical Expression ( square root of 7- square root of 5)/( square root of 12+ square root of 7)

Solution:

Step:1

Transform the denominator.

Step:1.1

Express $12$ as a product of its prime factors, $2^2\cdot 3$.

Step:1.1.1

Extract $4$ from $12$ to get $\frac{\sqrt{7}-\sqrt{5}}{\sqrt{4\cdot 3}+\sqrt{7}}$.

Step:1.1.2

Represent $4$ as $2^2$ to obtain $\frac{\sqrt{7}-\sqrt{5}}{\sqrt{2^2\cdot 3}+\sqrt{7}}$.

Step:1.2

Extract the square root of perfect squares from under the radical, yielding $\frac{\sqrt{7}-\sqrt{5}}{2\sqrt{3}+\sqrt{7}}$.

Step:1.3

Recognize that the absolute value of $2$ is $2$, simplifying to $\frac{\sqrt{7}-\sqrt{5}}{2\sqrt{3}+\sqrt{7}}$.

Step:2

Multiply the expression by the conjugate of the denominator $\frac{2\sqrt{3}-\sqrt{7}}{2\sqrt{3}-\sqrt{7}}$.

Step:3

Multiply the numerators and denominators to get $\frac{(\sqrt{7}-\sqrt{5})(2\sqrt{3}-\sqrt{7})}{(2\sqrt{3}+\sqrt{7})(2\sqrt{3}-\sqrt{7})}$.

Step:4

Apply the FOIL method to the denominator to expand it as $\frac{(\sqrt{7}-\sqrt{5})(2\sqrt{3}-\sqrt{7})}{4(\sqrt{3}{)}^2-(\sqrt{7}{)}^2}$.

Step:5

Simplify the denominator to get $\frac{(\sqrt{7}-\sqrt{5})(2\sqrt{3}-\sqrt{7})}{5}$.

Step:6

Expand the numerator using the FOIL method.

Step:6.1

Distribute $\sqrt{7}(2\sqrt{3}-\sqrt{7})-\sqrt{5}(2\sqrt{3}-\sqrt{7})$ over $5$.

Step:6.2

Continue distribution to get $\frac{\sqrt{7}(2\sqrt{3})-\sqrt{7}(\sqrt{7})-\sqrt{5}(2\sqrt{3})+\sqrt{5}(\sqrt{7})}{5}$.

Step:7

Simplify each term in the numerator.

Step:7.1

Multiply $\sqrt{7}(2\sqrt{3})$.

Step:7.1.1

Apply the product rule for radicals to get $\frac{2\sqrt{21}-\sqrt{7}(\sqrt{7})-\sqrt{5}(2\sqrt{3})+\sqrt{5}(\sqrt{7})}{5}$.

Step:7.1.2

Multiply $7$ by $3$ to obtain $\frac{2\sqrt{21}-7-\sqrt{5}(2\sqrt{3})+\sqrt{5}(\sqrt{7})}{5}$.

Step:7.2

Multiply $\sqrt{7}(\sqrt{7})$.

Step:7.2.1

Raise $\sqrt{7}$ to the power of $2$ to get $\frac{2\sqrt{21}-7-\sqrt{5}(2\sqrt{3})+\sqrt{5}(\sqrt{7})}{5}$.

Step:7.3

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

Step:7.3.1

Apply the power rule to rewrite ${\sqrt{7}}^2$ as $7$.

Step:7.3.2

Evaluate the exponent to get $\frac{2\sqrt{21}-7-\sqrt{5}(2\sqrt{3})+\sqrt{5}(\sqrt{7})}{5}$.

Step:7.4

Multiply $-\sqrt{5}(2\sqrt{3})$.

Step:7.4.1

Apply the product rule for radicals to get $\frac{2\sqrt{21}-7-2\sqrt{15}+\sqrt{5}(\sqrt{7})}{5}$.

Step:7.5

Multiply $\sqrt{5}(\sqrt{7})$.

Step:7.5.1

Apply the product rule for radicals to get $\frac{2\sqrt{21}-7-2\sqrt{15}+\sqrt{35}}{5}$.

Step:8

The final result can be presented in various forms.

Exact Form: $\frac{2\sqrt{21}-7-2\sqrt{15}+\sqrt{35}}{5}$ Decimal Form: Approximately $0.06705289\dots$

Knowledge Notes:

1. Radical Simplification: Simplifying radical expressions involves identifying and extracting perfect squares, cubes, etc., from under the radical sign.

2. Conjugate Multiplication: To rationalize a denominator containing radicals, multiply the numerator and denominator by the conjugate of the denominator.

3. FOIL Method: Stands for First, Outer, Inner, Last. It is a technique for expanding two binomials.

4. Product Rule for Radicals: $\sqrt{a}\cdot \sqrt{b}=\sqrt{ab}$.

5. Power Rule: For any non-negative real number $a$, $(\sqrt{a}{)}^2=a$.

6. Absolute Value: The absolute value of a number is its distance from zero on the number line, denoted as $|a|$ for a number $a$.

7. Rationalizing the Denominator: The process of eliminating radicals from the denominator of a fraction.