Solving Complex Equations and Their Applications in Mathematics
Mathematics is a discipline that often presents us with complex equations to solve, which test our problem-solving skills and understanding of fundamental concepts. One such intriguing problem involves the equation X1/X 7. This article will guide you through the steps to solve this equation and provide a detailed explanation of the mathematical processes involved.
Introduction to the Problem
The problem under discussion is to determine the value of the expression Xsqrt{X}/(Xsqrt{X}) given that X1/X 7. This problem involves algebraic manipulation and the application of various mathematical identities to arrive at the final solution.
Step-by-Step Solution
Step 1 - Define and Simplify the Expression
We start by defining Y sqrt{X}.
Step 2 - Substitute and Rearrange
Since X Y^2, we can substitute this into the given equation:
Y^2 * 1/Y^2 7
This simplifies to:
1 7 - 2
Let z Y / (1/Y) thus Y^2 * 1/Y^2 z^2 - 2. Setting this equal to 7 gives us:
z^2 - 2 7
Solving for z gives:
z ±3
Considering that Y sqrt{X} must be non-negative, we take z 3.
Step 3 - Further Simplification
We then need to find the value of Y^3 / (Y^3). By letting Y w^2, we can express the given expression in terms of w as follows:
w^3 (w / 1/w)^3 - 3 * (w / 1/w)
Substituting w / 1/w sqrt{5} we get:
w^3 / (1/w^3) (sqrt{5})^3 - 3(sqrt{5}) 2sqrt{5}
Therefore, the final solution to the equation is:
boxed{2sqrt{5}}
Additional Methods for Problem Solving
The problem can also be solved using hyperbolic functions and exponential expressions. For instance, using the hyperbolic cosine function:
x1/x 2cosh(ln(x)) 7
From this, we can determine x exp(cosh^(-1)(7/2)), and therefore:
xsqrt{x} exp(3/2cosh^(-1)(7/2))
And:
1/xsqrt{x} exp(-3/2cosh^(-1)(7/2))
The final expression evaluated by Wolfram Alpha is:
xsqrt{x} * 1/xsqrt{x} 2cosh(3/2cosh^(-1)(7/2)) 18
Conclusion
The process of solving complex equations, such as X1/X 7, involves a series of algebraic manipulations and the application of various mathematical identities. By following a systematic approach, we can determine the solution to the problem and gain a deeper understanding of the underlying mathematical concepts.