Simplifying Exponential Expressions A Step-by-Step Guide To C = X^{16^4 - 8^{0.5}} ÷ ((x^(b+10)^(1/3) * X^(1/6)) / (x^(b-2)^(1/4) * X^(3-b)^(1/12)))^(24/(2b+45))

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Unveiling the Intricacies of Exponential Expressions

At the heart of mathematical exploration lies the quest to simplify complex expressions. Our journey today takes us through the realm of exponents and radicals, where we aim to unravel the intricacies of the expression C = x{164 - 8^{0.5}} ÷ ( (x(b+10)(1/3) * x^(1/6)) / (x(b-2)(1/4) * x(3-b)(1/12)) )^(24/(2b+45)). This formidable equation, a tapestry of exponents, radicals, and variables, presents a fascinating challenge. By employing the fundamental principles of exponents and algebraic manipulation, we shall embark on a step-by-step simplification process, ultimately revealing the expression's underlying structure and meaning.

Delving into the Foundation: Exponents and Radicals

Before we embark on the simplification process, let's take a moment to solidify our understanding of the core concepts: exponents and radicals. An exponent signifies the number of times a base is multiplied by itself. For instance, in the expression x^n, 'x' is the base, and 'n' is the exponent, indicating that 'x' is multiplied by itself 'n' times. Radicals, on the other hand, represent the inverse operation of exponents. The nth root of a number 'x' is denoted as x^(1/n), signifying the value that, when raised to the power of 'n', equals 'x'. These fundamental concepts form the bedrock of our simplification journey.

Navigating the Order of Operations

As we delve into the intricacies of our expression, it's crucial to adhere to the order of operations, a guiding principle that ensures consistency and accuracy in mathematical calculations. The acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) serves as our roadmap, dictating the sequence in which operations are performed. By meticulously following this order, we can navigate the complexities of the expression and arrive at a simplified form.

Step-by-Step Simplification Process

1. Simplifying the Initial Exponent: 16^4 - 8^{0.5}

Our simplification journey commences with the exponent in the numerator: 16^4 - 8^{0.5}. Let's break this down step by step:

  • 16^4: This represents 16 multiplied by itself four times, resulting in 65,536.
  • 8^{0.5}: This is equivalent to the square root of 8, which can be expressed as √(8) = 2√(2).

Therefore, 16^4 - 8^{0.5} simplifies to 65,536 - 2√(2).

2. Simplifying the Denominator's Innermost Expressions

The denominator of our expression presents a more intricate challenge, requiring us to simplify a series of radicals and exponents. Let's dissect this part step by step:

  • (x(b+10))(1/3): This represents the cube root of x^(b+10), which can be written as x^((b+10)/3).
  • x^(1/6): This represents the sixth root of x.
  • (x(b-2))(1/4): This represents the fourth root of x^(b-2), which can be written as x^((b-2)/4).
  • (x(3-b))(1/12): This represents the twelfth root of x^(3-b), which can be written as x^((3-b)/12).

3. Combining Terms within the Parentheses

Now, let's combine the terms within the parentheses in the denominator:

(x^((b+10)/3) * x^(1/6)) / (x^((b-2)/4) * x^((3-b)/12))

To simplify this, we apply the rule of exponents that states x^m * x^n = x^(m+n) and x^m / x^n = x^(m-n):

  • Numerator: x^((b+10)/3) * x^(1/6) = x^((b+10)/3 + 1/6)
  • Denominator: x^((b-2)/4) * x^((3-b)/12) = x^((b-2)/4 + (3-b)/12)

Let's find a common denominator to add the exponents:

  • (b+10)/3 + 1/6 = (2(b+10) + 1)/6 = (2b + 21)/6
  • (b-2)/4 + (3-b)/12 = (3(b-2) + (3-b))/12 = (3b - 6 + 3 - b)/12 = (2b - 3)/12

Therefore, the expression within the parentheses simplifies to:

x^((2b+21)/6) / x^((2b-3)/12) = x^((2b+21)/6 - (2b-3)/12)

Again, let's find a common denominator to subtract the exponents:

(2b+21)/6 - (2b-3)/12 = (2(2b+21) - (2b-3))/12 = (4b + 42 - 2b + 3)/12 = (2b + 45)/12

So, the expression within the parentheses now simplifies to:

x^((2b+45)/12)

4. Applying the Outer Exponent: 24/(2b+45)

Next, we need to apply the outer exponent of 24/(2b+45) to the simplified expression within the parentheses:

(x((2b+45)/12))(24/(2b+45))

Using the rule of exponents that states (xm)n = x^(m*n), we get:

x^(((2b+45)/12) * (24/(2b+45))) = x^(24/12) = x^2

5. Final Simplification: C = x^(65,536 - 2√(2)) ÷ x^2

Finally, we can substitute the simplified expressions back into the original equation:

C = x^(65,536 - 2√(2)) ÷ x^2

Using the rule of exponents that states x^m / x^n = x^(m-n), we get:

C = x^(65,536 - 2√(2) - 2)

Therefore, the simplified expression is:

C = x^(65,534 - 2√(2))

Conclusion: A Triumph of Simplification

Through a meticulous step-by-step process, we have successfully simplified the complex expression C = x{164 - 8^{0.5}} ÷ ( (x(b+10)(1/3) * x^(1/6)) / (x(b-2)(1/4) * x(3-b)(1/12)) )^(24/(2b+45)) to its elegant form: C = x^(65,534 - 2√(2)). This journey has not only demonstrated the power of exponents and radicals but also highlighted the importance of adhering to the order of operations and employing algebraic manipulation techniques. The simplified expression offers a clearer understanding of the relationship between the variables and constants involved, showcasing the beauty and elegance of mathematical simplification.

In essence, this simplification process exemplifies the core principles of mathematics: transforming complex problems into manageable steps, applying fundamental rules and theorems, and ultimately arriving at a concise and insightful solution. The ability to simplify expressions is a cornerstone of mathematical proficiency, enabling us to tackle intricate problems and gain deeper insights into the world around us.