Understand the problem: Understand the problem. We need to simplify the expression tan^(-1)((1)/(8x+9)). This is the inverse tangent (also known as arctan) of the fraction (1)/(8x+9).Recognize properties: Recognize the properties of the inverse tangent function. The inverse tangent function, tan^(-1)(x), does not have algebraic simplifications like the ones we use for powers or roots. It is a trigonometric function that gives us an angle whose tangent is x. Therefore, the expression tan^(-1)((1)/(8x+9)) is already in its simplest form as a function of x.Check for simplifications: Check for any possible simplifications. Since the expression inside the inverse tangent function is a simple fraction and cannot be simplified further, and there are no trigonometric identities that apply to inverse tangent functions with a variable inside, we conclude that the expression tan^(-1)((1)/(8x+9)) is already in its simplest form.

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$\tan ^{-1}\left(\frac{1}{8 x+9}\right)$

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Algebra 1

Evaluate integers raised to rational exponents

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\tan ^{-1}\left(\frac{1}{8 x+9}\right)

Understand the problem: Understand the problem. $\newline$ We need to simplify the expression $\tan^{-1}\left(\frac{1}{8x+9}\right)$ . This is the inverse tangent (also known as arctan) of the fraction $\frac{1}{8x+9}$ .
Recognize properties: Recognize the properties of the inverse tangent function. $\newline$ The inverse tangent function, $\tan^{-1}(x)$ , does not have algebraic simplifications like the ones we use for powers or roots. It is a trigonometric function that gives us an angle whose tangent is $x$ . Therefore, the expression $\tan^{-1}\left(\frac{1}{8x+9}\right)$ is already in its simplest form as a function of $x$ .
Check for simplifications: Check for any possible simplifications. Since the expression inside the inverse tangent function is a simple fraction and cannot be simplified further, and there are no trigonometric identities that apply to inverse tangent functions with a variable inside, we conclude that the expression $\tan^{-1}\left(\frac{1}{8x+9}\right)$ is already in its simplest form.