Solving the Equation: 5 - √3 / 5√3 ab√3 for a and b

The problem statement given is 5 - √3 / 5√3 ab√3. This involves the manipulation of fractions, square roots, and algebraic simplification. Let's break down the equation step by step to find the values of a and b. We will use LaTeX for mathematical expressions to ensure clarity and precision.

Solving the Equation

To simplify the left-hand side (LHS) of the equation, we start with the fraction:

[text{LHS} frac{5 - sqrt{3}}{5sqrt{3}}]

Our goal is to simplify the numerator in the fraction. To do this, we will rationalize the fraction by multiplying both the numerator and the denominator by the conjugate of the denominator:

[frac{5 - sqrt{3}}{5sqrt{3}} times frac{5 - sqrt{3}}{5 - sqrt{3}} frac{(5 - sqrt{3})^2}{5^2 - (sqrt{3})^2}]

Simplifying the denominator, we get:

[5^2 - (sqrt{3})^2 25 - 3 22]

And the numerator:

[(5 - sqrt{3})^2 25 - 2 times 5 times sqrt{3} (sqrt{3})^2 25 - 10sqrt{3} 3 28 - 10sqrt{3}]

So the equation becomes:

[frac{28 - 10sqrt{3}}{22} frac{28}{22} - frac{10sqrt{3}}{22} frac{14}{11} - frac{5sqrt{3}}{11}]

The right-hand side (RHS) is given as (asqrt{3} b). By comparing the simplified LHS to the RHS, we can identify the values of (a) and (b).

[text{LHS} frac{14}{11} - frac{5sqrt{3}}{11} asqrt{3} b]

Hence, we can conclude that:

[a -frac{5}{11}, quad b frac{14}{11}]

This solution provides the values of (a) and (b) that satisfy the given equation.

Final Answer

[a -frac{5}{11}, quad b frac{14}{11}]

Conclusion

Understanding and solving such equations involves familiarizing oneself with algebraic manipulations, rationalization, and recognizing patterns in expressions involving square roots. This problem is a classic example of how fractions and square roots can be combined and simplified to find unknown variables.

Keywords

Math Equations, Algebra, Square Root Simplification

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