Simplify ln(root(3)(e^(4))) Answer:

Understand the expression: Understand the expression ln(root(3)(e^(4))). This expression involves the natural logarithm of the cube root of e raised to the power of 4. The cube root can be expressed as a fractional exponent.Convert to fractional exponent: Convert the cube root to a fractional exponent. The cube root of e^(4) is the same as e^(4/3). So, ln(root(3)(e^(4))) becomes ln(e^(4/3)).Apply logarithm power rule: Apply the logarithm power rule. The power rule of logarithms states that ln(a^b) = b * ln(a). We apply this rule to simplify ln(e^(4/3)) to (4/3) * ln(e).Simplify using ln(e)=1: Simplify the expression using the fact that ln(e) = 1. Since the natural logarithm of e is 1, the expression (4/3) * ln(e) simplifies to (4/3) * 1.Perform multiplication: Perform the multiplication to find the final answer. (4/3) * 1 equals 4/3. Therefore, the simplified form of the expression is 4/3.

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Algebra 2

Simplify variable expressions using properties

Full solution

Q. Simplify

\ln \left(\sqrt[3]{e^{4}}\right)

\newline

Answer:

Understand the expression: Understand the expression $\ln(\sqrt[3]{e^{4}})$ . This expression involves the natural logarithm of the cube root of $e$ raised to the power of $4$ . The cube root can be expressed as a fractional exponent.
Convert to fractional exponent: Convert the cube root to a fractional exponent. The cube root of $e^{4}$ is the same as $e^{\frac{4}{3}}$ . So, $\ln(\sqrt[3]{e^{4}})$ becomes $\ln(e^{\frac{4}{3}})$ .
Apply logarithm power rule: Apply the logarithm power rule. $\newline$ The power rule of logarithms states that $\ln(a^b) = b \cdot \ln(a)$ . We apply this rule to simplify $\ln(e^{\frac{4}{3}})$ to $\frac{4}{3} \cdot \ln(e)$ .
Simplify using $\ln(e)=1$ : Simplify the expression using the fact that $\ln(e) = 1$ . $\newline$ Since the natural logarithm of $e$ is $1$ , the expression $(4/3) \times \ln(e)$ simplifies to $(4/3) \times 1$ .
Perform multiplication: Perform the multiplication to find the final answer. $\newline$ $(\frac{4}{3}) \times 1$ equals $\frac{4}{3}$ . Therefore, the simplified form of the expression is $\frac{4}{3}$ .