Rewrite the expressions in Exercises 5–10 in terms of exponentials and simplify the results as much as you can.
9. (sinh(x)+cosh(x))⁴

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Recall the definitions of hyperbolic sine and cosine in terms of exponentials: \(\sinh(x) = \frac{e^{x} - e^{-x}}{2}\) and \(\cosh(x) = \frac{e^{x} + e^{-x}}{2}\).

Add \(\sinh(x)\) and \(\cosh(x)\) using their exponential forms: \(\sinh(x) + \cosh(x) = \frac{e^{x} - e^{-x}}{2} + \frac{e^{x} + e^{-x}}{2}\).

Combine the fractions since they have the same denominator: \(\frac{e^{x} - e^{-x} + e^{x} + e^{-x}}{2} = \frac{2e^{x}}{2} = e^{x}\).

Rewrite the original expression \((\sinh(x) + \cosh(x))^{4}\) as \((e^{x})^{4}\) using the simplification from the previous step.

Apply the exponent rule \((e^{x})^{4} = e^{4x}\) to express the final simplified form in terms of exponentials.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Hyperbolic Functions and Their Definitions

Hyperbolic sine (sinh) and cosine (cosh) are defined using exponential functions: sinh(x) = (e^x - e^(-x))/2 and cosh(x) = (e^x + e^(-x))/2. Understanding these definitions allows rewriting expressions involving sinh and cosh in terms of exponentials.

Algebraic Simplification of Exponential Expressions

After rewriting hyperbolic functions as exponentials, simplifying involves combining like terms, factoring, and applying exponent rules. Mastery of these algebraic techniques is essential to reduce the expression to its simplest form.

Properties of Exponents and Binomial Expansion

Raising a sum to a power, such as (sinh(x) + cosh(x))^4, often requires using binomial expansion or recognizing patterns in exponentials. Understanding exponent properties helps in expanding and simplifying the expression efficiently.

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