Simplify the Radical Expression ( square root of 7- square root of 5)/( square root of 12+ square root of 7)
The question asks you to simplify a given algebraic expression that involves radicals (square roots in particular). The expression you need to simplify is a rational expression, comprising a numerator and a denominator, each containing a combination of square root terms. Specifically, the numerator is the difference between the square root of 7 and the square root of 5, while the denominator is the sum of the square root of 12 and the square root of 7. Simplifying the expression generally involves rationalizing the denominator (if necessary) and combining like terms in a way that results in the simplest form of the expression with no square roots in the denominator.
$\frac{\sqrt{7} - \sqrt{5}}{\sqrt{12} + \sqrt{7}}$
Answer
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 $0.06705289 \ldots$
Knowledge Notes:
Radical Simplification: Simplifying radical expressions involves identifying and extracting perfect squares, cubes, etc., from under the radical sign.
Conjugate Multiplication: To rationalize a denominator containing radicals, multiply the numerator and denominator by the conjugate of the denominator.
FOIL Method: Stands for First, Outer, Inner, Last. It is a technique for expanding two binomials.
Product Rule for Radicals: $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$.
Power Rule: For any non-negative real number $a$, $(\sqrt{a})^2 = a$.
Absolute Value: The absolute value of a number is its distance from zero on the number line, denoted as $|a|$ for a number $a$.
Rationalizing the Denominator: The process of eliminating radicals from the denominator of a fraction.
