Problem 51 - College Algebra with Modeling and Visualization

Question

Write the expression in radical notation. Then evaluate the expression when the result is an integer. $$ 8^{-1 / 3} $$

Step-by-Step Solution

Answer

The expression in radical notation is \( \frac{1}{\sqrt[3]{8}} \), and its evaluated value is \( \frac{1}{2} \).

Step 1: Express as Radical Notation

To convert the expression \( 8^{-1/3} \) into radical form, recognize that the exponent \(-1/3\) indicates both a reciprocal and a cube root. A negative exponent means we take the reciprocal, and the \(1/3\) exponent indicates a cube root. So, \( 8^{-1/3} \) can be expressed as \( \frac{1}{8^{1/3}} \). This simplifies further to \( \frac{1}{\sqrt[3]{8}} \).

Step 2: Evaluate the Cube Root

To evaluate \( \sqrt[3]{8} \), determine what number, when cubed, equals 8. Since \( 2^3 = 8 \), we find that \( \sqrt[3]{8} = 2 \).

Step 3: Apply the Reciprocal

Given that \( \sqrt[3]{8} = 2 \), apply the reciprocal as indicated by the negative exponent. Therefore, \( 8^{-1/3} = \frac{1}{2} \).

Key Concepts

Negative Exponents

A negative exponent indicates the need for the reciprocal of a base raised to the positive version of that exponent. In simple terms, it tells us to "flip" the fraction. For example, when you see an expression such as \( a^{-n} \), you should rewrite it as \( \frac{1}{a^n} \).
Let's consider our problem, \( 8^{-1/3} \), the presence of the negative sign tells us we'll have to take the reciprocal.

• The base is 8.
• The reciprocal is \( \frac{1}{8} \).
• The exponent, \( -1/3 \), applies to \( 8 \) before considering its negative sign.

This concept is important, as it transforms the problem into a more manageable form, allowing us to work with positive exponents afterward.

Cube Roots

Cube roots are essential when dealing with exponents of \( 1/3 \), as they ask us to find the number which, when raised to the power of three, equals the original number. Mathematically, the cube root of a number \( x \) is written as \( \sqrt[3]{x} \).
In our example, \( 8^{-1/3} \) involves finding the cube root of 8:

• The cube root \( \sqrt[3]{8} \) asks, "What number multiplied by itself three times equals 8?"
• We know \( 2 \times 2 \times 2 = 8 \), hence \( \sqrt[3]{8} = 2 \).

Knowing how to find and use cube roots helps simplify expressions involving fractional exponents, especially when calculating other parts like reciprocals in our exercise.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations that represent a particular value or set of values. In exercises like this, expressions may require transformations using exponent rules and operations like roots.
For \( 8^{-1/3} \), the expression includes both exponents and roots. Decomposing the expression, we:

• Recognized \( 8^{-1/3} \) as \( \frac{1}{8^{1/3}} \).
• Calculated with the root \( \sqrt[3]{8} \), resulting in 2.
• Utilized the reciprocal due to the negative exponent, arriving at \( \frac{1}{2} \).

Understanding how to manipulate and analyze algebraic expressions is crucial for solving complex mathematical problems efficiently, allowing us to break them down into simpler, more manageable parts.