Problem 48 - College Algebra with Modeling and Visualization

Question

Write the expression in radical notation. Then evaluate the expression when the result is an integer. $$ 8^{4 / 3} $$

Step-by-Step Solution

Answer

The radical notation is \((\sqrt[3]{8})^4\), and the evaluated expression is 16.

Step 1: Understand the Expression

The expression given is in the form of an exponent and is written as \(8^{4/3}\). This implies a power and a root in the expression.

Step 2: Convert to Radical Notation

The expression \(8^{4/3}\) can be rewritten using radical notation. The denominator \(3\) indicates a cube root, and the numerator \(4\) is a power, so it becomes \((\sqrt[3]{8})^4\).

Step 3: Evaluate the Cube Root

First, evaluate the cube root: \(\sqrt[3]{8}\). Since \(8\) is \(2^3\), the cube root is \(2\). So we have \((2)^4\).

Step 4: Evaluate the Power

Now evaluate \((2)^4\). Calculating \(2^4 = 16\).

Step 5: Final Result

The expression \(8^{4/3}\) evaluates to \(16\), which is an integer value.

Key Concepts

Exponential Expressions

Exponential expressions are mathematical expressions involving numbers raised to powers. In simple terms, an exponential expression has a base and an exponent. The exponent tells us how many times the base is multiplied by itself. For example, in the expression \( 8^{4/3} \), \( 8 \) is the base, and \( \frac{4}{3} \) is the exponent.
This expression signifies that we take the base, \( 8 \), and apply both a root (denominator of the exponent) and a power (numerator of the exponent).

• Fractional exponents represent both roots and powers; the denominator is the type of root, and the numerator is the power.
• They provide a convenient way to express more complex operations in a compact form.

In this example, \( 8^{4/3} \) signifies identifying the cube root of \( 8 \) and then raising the result to the fourth power.

Radical Notation

Radical notation is another way of expressing the concept of roots. It uses the radical symbol \( \sqrt[]{} \) to denote roots of numbers, with a small index number to indicate the type of root.
When we convert an exponential expression like \( 8^{4/3} \) into radical notation, the fraction exponent becomes a root and power operation.
For instance:

• The denominator \( 3 \) in \( \frac{4}{3} \) tells us to find the cube root, represented as \( \sqrt[3]{} \).
• The numerator \( 4 \) indicates the power, shown outside the radical as \((\sqrt[3]{8})^4\).

This conversion process is vital to evaluate expressions accurately, especially when calculating precise values for non-integer exponents.

Integer Evaluation

Integer evaluation refers to calculating the value of an expression where the result is a whole number, with no fractions or decimals. In our exercise, the task is to evaluate \( 8^{4/3} \) in such a way.
The process consists of two main steps:

• First, compute the root, which is solving \( \sqrt[3]{8} \). Since \( 8 \) is \( 2^3 \), the cube root is easy to identify as \( 2 \).
• Next, calculate the power \( (2)^4 \), resulting in \( 16 \).

The end result, \( 16 \), is an integer, fulfilling the requirements of the problem. Integer evaluation is essential in algebra to confirm that operations have been carried out correctly and that results are straightforward whole numbers.