Simplify (2+ square root of 14)/( square root of 7+ square root of 2)
The problem asks for a simplification of a given rational expression. Specifically, the expression to be simplified is a quotient, where the numerator is the sum of 2 and the square root of 14, and the denominator is the sum of the square root of 7 and the square root of 2. The challenge is to rationalize the denominator and rewrite the expression in a simpler form, potentially making it easier to evaluate or use in further calculations. Rationalizing the denominator typically involves eliminating any square roots from the bottom of the fraction by multiplying the numerator and denominator by a suitable expression that will achieve this goal without changing the value of the original expression.
$\frac{2 + \sqrt{14}}{\sqrt{7} + \sqrt{2}}$
Answer
Solution:
Step:1
Rationalize the denominator of $\frac{2 + \sqrt{14}}{\sqrt{7} + \sqrt{2}}$ by multiplying by the conjugate $\frac{\sqrt{7} - \sqrt{2}}{\sqrt{7} - \sqrt{2}}$.
$$\frac{2 + \sqrt{14}}{\sqrt{7} + \sqrt{2}} \cdot \frac{\sqrt{7} - \sqrt{2}}{\sqrt{7} - \sqrt{2}}$$
Step:2
Combine the fractions into a single expression.
Step:2.1
Multiply the numerators and denominators.
$$\frac{(2 + \sqrt{14})(\sqrt{7} - \sqrt{2})}{(\sqrt{7} + \sqrt{2})(\sqrt{7} - \sqrt{2})}$$
Step:2.2
Apply the difference of squares to the denominator.
$$\frac{(2 + \sqrt{14})(\sqrt{7} - \sqrt{2})}{7 - 2}$$
Step:2.3
Simplify the denominator.
$$\frac{(2 + \sqrt{14})(\sqrt{7} - \sqrt{2})}{5}$$
Step:3
Expand and simplify the numerator.
Step:3.1
Use the distributive property (FOIL) to expand the numerator.
Step:3.1.1
Distribute the terms.
$$\frac{2(\sqrt{7} - \sqrt{2}) + \sqrt{14}(\sqrt{7} - \sqrt{2})}{5}$$
Step:3.1.2
Continue distributing.
$$\frac{2\sqrt{7} - 2\sqrt{2} + \sqrt{14}\sqrt{7} - \sqrt{14}\sqrt{2}}{5}$$
Step:3.2
Combine like terms and simplify.
Step:3.2.1
Simplify each term.
Step:3.2.1.1
Multiply and simplify.
$$\frac{2\sqrt{7} - 2\sqrt{2} + 7\sqrt{2} - \sqrt{28}}{5}$$
Step:3.2.1.2
Factor and simplify the radical.
$$\frac{2\sqrt{7} - 2\sqrt{2} + 7\sqrt{2} - 2\sqrt{7}}{5}$$
Step:3.2.2
Subtract like terms.
$$\frac{0 - 2\sqrt{2} + 7\sqrt{2}}{5}$$
Step:3.2.3
Combine the remaining terms.
$$\frac{5\sqrt{2}}{5}$$
Step:4
Cancel out the common factors.
Step:4.1
Reduce the fraction.
$$\frac{\cancel{5}\sqrt{2}}{\cancel{5}}$$
Step:4.2
Final simplification.
$$\sqrt{2}$$
Step:5
Present the result in various forms.
Exact Form: $\sqrt{2}$ Decimal Form: $1.41421356\ldots$
Knowledge Notes:
Rationalizing the Denominator: This involves multiplying the numerator and denominator by the conjugate of the denominator to eliminate the radical from the denominator.
Difference of Squares: A technique used to simplify expressions like $(a+b)(a-b)$ which equals $a^2 - b^2$.
Distributive Property (FOIL): Used to expand expressions by multiplying each term in the first expression by each term in the second expression.
Combining Like Terms: This is the process of adding or subtracting terms that have the same variable raised to the same power.
Simplifying Radicals: Involves expressing the radical in its simplest form, often by factoring out squares from under the radical sign.
Reducing Fractions: The process of dividing the numerator and denominator by their greatest common factor to simplify the fraction.
Conjugate: The conjugate of a binomial $a + b$ is $a - b$. Multiplying a binomial by its conjugate results in a difference of squares, which is a technique often used to rationalize denominators.
