Simplify (-2+ square root of 3)/( square root of 3- square root of 15)
The problem presented involves simplifying a given algebraic fraction. The fraction consists of a numerator, which is the expression (-2 plus the square root of 3), and a denominator, which is the expression (square root of 3 minus the square root of 15). The square root symbol (√) denotes the principal square root function. The task requires using algebraic methods to simplify the fraction to its simplest form, which might involve rationalizing the denominator by eliminating the square roots in the denominator through multiplication by a suitable conjugate pair or other algebraic manipulations.
$\frac{- 2 + \sqrt{3}}{\sqrt{3} - \sqrt{15}}$
Answer
Solution:
Simplification Process
Step 1: Rationalize the Denominator
To rationalize the denominator, multiply the fraction by a form of 1 that will eliminate the radicals in the denominator. In this case, multiply by the conjugate of the denominator:
$$\frac{-2 + \sqrt{3}}{\sqrt{3} - \sqrt{15}} \times \frac{\sqrt{3} + \sqrt{15}}{\sqrt{3} + \sqrt{15}}$$
Step 2: Combine the Fractions
Step 2.1: Multiply Numerator and Denominator
Multiply the numerators and denominators:
$$\frac{(-2 + \sqrt{3})(\sqrt{3} + \sqrt{15})}{(\sqrt{3} - \sqrt{15})(\sqrt{3} + \sqrt{15})}$$
Step 2.2: Expand the Denominator Using Difference of Squares
Apply the difference of squares formula to the denominator:
$$\frac{(-2 + \sqrt{3})(\sqrt{3} + \sqrt{15})}{(\sqrt{3})^2 - (\sqrt{15})^2}$$
Step 2.3: Simplify the Denominator
Simplify the denominator by subtracting the squares:
$$\frac{(-2 + \sqrt{3})(\sqrt{3} + \sqrt{15})}{-12}$$
Step 3: Expand the Numerator Using the Distributive Property
Step 3.1: Distribute Each Term in the First Binomial Across the Second
Multiply each term in the first binomial by each term in the second binomial:
$$\frac{-2(\sqrt{3} + \sqrt{15}) + \sqrt{3}(\sqrt{3} + \sqrt{15})}{-12}$$
Step 3.2: Continue Distributing
Distribute the terms:
$$\frac{-2\sqrt{3} - 2\sqrt{15} + \sqrt{3}\sqrt{3} + \sqrt{3}\sqrt{15}}{-12}$$
Step 4: Simplify the Numerator
Step 4.1: Simplify Each Term in the Numerator
Combine and simplify terms:
$$\frac{-2\sqrt{3} - 2\sqrt{15} + 3 + \sqrt{45}}{-12}$$
Step 4.2: Factor Out Common Terms and Simplify the Expression
Step 4.2.1: Factor Out a Negative One from the Numerator
Factor out -1 to simplify the fraction:
$$-\frac{2\sqrt{3} + 2\sqrt{15} - 3 - 3\sqrt{5}}{12}$$
Step 4.2.2: Simplify the Negative Signs
Simplify the negative signs and the fraction:
$$\frac{2\sqrt{3} + 2\sqrt{15} - 3 - 3\sqrt{5}}{12}$$
Step 5: Present the Result in Different Forms
The simplified expression can be presented in exact form and decimal form:
Exact Form: $$\frac{2\sqrt{3} + 2\sqrt{15} - 3 - 3\sqrt{5}}{12}$$ Decimal Form: Approximately $0.12515536 \ldots$
Knowledge Notes:
To solve the given problem, we used several algebraic techniques:
Rationalizing the Denominator: This involves multiplying the numerator and denominator by the conjugate of the denominator to eliminate radicals from the denominator.
Difference of Squares: This is a pattern used to simplify expressions of the form $a^2 - b^2$ into $(a + b)(a - b)$.
Distributive Property (FOIL Method): This property is used to expand expressions by multiplying each term in one binomial by each term in another.
Combining Like Terms: This involves adding or subtracting terms that have the same variable raised to the same power.
Simplifying Radicals: This includes operations like combining radicals using the product rule, factoring numbers inside the radical to simplify them, and pulling out perfect squares from under the radical.
Factoring: This is the process of taking out common factors from terms to simplify expressions.
By applying these techniques systematically, we can simplify complex algebraic expressions and rationalize denominators that contain radicals.
