na jednak eksponent. $\newline$ $2$ . Odredi vrijednost od $x$ u binomu $\left((\sqrt{x})^{\frac{1}{\log x+1}}+\sqrt[12]{x}\right)^{6}$ čiji je četvrti član razvoja binoma jednak $200$ .

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Evaluate rational exponents

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Q. na jednak eksponent.

\newline

2

. Odredi vrijednost od

x

u binomu

\left((\sqrt{x})^{\frac{1}{\log x+1}}+\sqrt[12]{x}\right)^{6}

čiji je četvrti član razvoja binoma jednak

200

Recognize Binomial Expansion Form: Recognize the general form of the binomial expansion $(a+b)^n$ and identify the fourth term using the binomial theorem.
Calculate Fourth Term Formula: The fourth term of the expansion is given by the formula $T_4 = \binom{n}{3} \cdot a^{(n-3)} \cdot b^3$ , where $\binom{n}{3}$ is the binomial coefficient for the fourth term.
Substitute Given Values: Substitute the given values into the formula: $T_4 = 6C_3 \times (\sqrt{x})^{(\frac{1}{\log(x)+1})^{6-3}} \times (\sqrt[12]{x})^3$ .
Calculate Binomial Coefficient: Calculate the binomial coefficient $6C3 = \frac{6!}{3! \times (6-3)!} = 20$ .
Simplify Powers in Terms: Simplify the powers in the terms: $(\sqrt{x})^{(\frac{1}{\log(x)+1})^3}$ becomes $x^{\frac{3}{2} * \frac{1}{\log(x)+1}}$ and $(\sqrt[12]{x})^3$ becomes $x^{\frac{3}{12}}$ or $x^{\frac{1}{4}}$ .
Set Up Equation: Set up the equation $20 \times x^{(3/2 \times 1/(\log(x)+1))} \times x^{(1/4)} = 200$ .
Combine X Terms: Combine the x terms by adding the exponents: $x^{\frac{3}{2} \cdot \frac{1}{\log(x)+1} + \frac{1}{4}} = \frac{200}{20}$ .
Simplify Right Side: Simplify the right side of the equation: $\frac{200}{20} = 10$ .
Isolate X: Now we have $x^{\frac{3}{2} \cdot \frac{1}{\log(x)+1} + \frac{1}{4}} = 10$ .
Approximate Solution: To solve for $x$ , we need to isolate it. However, we have $x$ in the exponent and as the base of a logarithm, which makes it a transcendental equation that's not straightforward to solve algebraically.
Approximate Solution: To solve for $x$ , we need to isolate it. However, we have $x$ in the exponent and as the base of a logarithm, which makes it a transcendental equation that's not straightforward to solve algebraically.We would typically use numerical methods or graphing to approximate the value of $x$ . Since we're looking for a more straightforward solution, we might have made a mistake in our previous steps.