Problem 49 - Intermediate Algebra

Question

Evaluate each expression without using a calculator. $$ \ln \sqrt[3]{e^{2}} $$

Step-by-Step Solution

Answer

The expression evaluates to \( \frac{2}{3} \).

Step 1: Identify the Inner Expression

The given expression is \( \ln \sqrt[3]{e^2} \). Notice that \( \sqrt[3]{e^2} \) represents the cube root of \( e^2 \). Our task will be to simplify this expression step by step.

Step 2: Convert Cube Root to Exponent

Recall that the cube root of a number is the same as raising that number to the power of \( \frac{1}{3} \). Therefore, \( \sqrt[3]{e^2} \) can be rewritten as \( (e^2)^{\frac{1}{3}} \).

Step 3: Apply Exponent Rule

Use the property of exponents that states \( (a^m)^n = a^{m \cdot n} \) to simplify \( (e^2)^{\frac{1}{3}} \). This becomes \( e^{2 \cdot \frac{1}{3}} = e^{\frac{2}{3}} \).

Step 4: Simplify with Natural Logarithm

Now apply the logarithmic rule \( \ln(a^b) = b \cdot \ln(a) \). For \( \ln(e^{\frac{2}{3}}) \), this results in \( \frac{2}{3} \cdot \ln(e) \).

Step 5: Evaluate Natural Logarithm

Recognize that \( \ln(e) = 1 \) because the natural logarithm of the base of the natural logarithm (\( e \)) is always 1. Thus, \( \frac{2}{3} \cdot 1 = \frac{2}{3} \).

Key Concepts

Cube Root

A cube root is a special kind of root that can be thought of as the inverse operation of cubing a number. When you take the cube root of a number, you're looking for a number which, when multiplied by itself three times, gives the original number. For example, the cube root of 8 is 2 because \( 2 \times 2 \times 2 = 8 \). The cube root is represented using the radical symbol with a small three in front, like this: \( \sqrt[3]{x} \).

In equation form, if you have \( \sqrt[3]{x} = y \), it means that \( y^3 = x \). This makes cube roots particularly useful when dealing with powers or exponents in expressions, allowing you to transform the operation to a multiplication of fractionally powered exponents. For instance, the cube root of \( e^2 \) can be rewritten as \( (e^2)^{\frac{1}{3}} \). This is an essential concept in solving complex equations involving roots and exponential functions.

Exponent Rules

Exponent rules are fundamental in simplifying and solving algebraic expressions involving powers. One of the most important rules deals with powers raised to powers. Given an expression \((a^m)^n\), the steP is to multiply the exponents: \( a^{m \cdot n} \). This makes simplifying expressions much more straightforward.

For example, in the context of our exercise, we have \((e^2)^{\frac{1}{3}}\). Applying the multiplication rule for exponents, this expression becomes \( e^{2 \cdot \frac{1}{3}} \) or simply \( e^{\frac{2}{3}} \).
Another important aspect of exponent rules is understanding how exponents can help simplify expressions before applying additional operations, such as logarithms. They are essential in transforming expressions into more workable forms, especially when involving roots or fractional exponents.

Logarithmic Properties

Logarithms are the inverse operations of exponentiation, which makes them particularly useful for solving equations where exponents are involved. The natural logarithm, denoted as \( \ln \), has special properties due to its base \( e \), an important mathematical constant approximately equal to 2.718.

One important property of logarithms is: \( \ln(a^b) = b \cdot \ln(a) \). This property allows us to simplify the logarithm of power by pulling the exponent outside the log function as a multiplier. In the given problem, we use it to simplify \( \ln(e^{\frac{2}{3}}) \) to \( \frac{2}{3} \cdot \ln(e) \).
Knowing that \( \ln(e) = 1 \) because \( e \) is the base of natural logs, makes the expression even simpler, resulting in \( \frac{2}{3} \).

• Understanding this property helps in breaking down complex logarithmic expressions.
• It's especially useful in calculus and higher-level mathematics where logs are frequently applied.