Problem 85 - Algebra and Trigonometry

Question

Evaluate each expression without using a calculator. $$ 8^{\frac{1}{3}} $$

Step-by-Step Solution

Answer

The value of the expression \(8^{1/3}\) is 2.

Step 1: Understand the Fractional Exponent

The number presented as an exponent (1/3) is a fraction. Fractional exponents represent roots. In this case, the denominator of the fraction '3' represents the type of root we will be finding, which is a cube root.

Step 2: Apply the Fractional Exponent

The given expression is \(8^{1/3}\). The exponent of 1/3 can be re-written as the cube root. Thus, \(8^{1/3}\) is the cube root of 8.

Step 3: Simplify the Expression

The cube root of 8 equals 2 because \(2^3 = 8\).

Key Concepts

Evaluating Exponential Expressions

Understanding how to work with exponential expressions is vital in mathematics, and evaluating them involves applying specific rules of exponents systematically. An exponential expression is one where a number, known as the base, is raised to the power of an exponent. When the exponent is a fraction, it indicates a root. For instance, when evaluating an expression like \(8^{\frac{1}{3}}\), you are being asked to find the cube root of 8.

The process typically can be broken down into identifying the base and the exponent, determining the type of root based on the denominator of the exponent, and then computing the value. In this case, the base is 8, and since the exponent is \(\frac{1}{3}\), you are looking for the number that, when raised to the power of 3, gives you 8. Thorough comprehension and correct application of these rules allow for the easy and accurate evaluation of exponential expressions.

Cube Roots

Cube roots are a type of radical expression and are the inverse operation of cubing a number. This means that the cube root of a given number is a value that, when multiplied by itself three times, yields the original number. For example, considering the cube root of 27, which is expressed as \(\sqrt[3]{27}\), you are searching for a number that, when cubed, equals 27. The answer is 3, since \(3^3 = 27\).

Visualizing Cube Roots

Imagine a cube where each edge has the same length. The volume of the cube is calculated by raising the edge length to the power of three. The cube root essentially asks in reverse: given a volume, what is the edge length of this cube? It's like packing the same amount of substance into a cube and figuring out how long each edge is. This visual approach can sometimes make understanding cube roots more tangible.

Simplifying Exponential Expressions

Simplifying exponential expressions can range from straightforward to complex, depending on the numbers and variables involved. When working with fractional exponents, simplifying means finding the root of the base as indicated by the denominator of the exponent. It's also critical to consider the numerator of the exponent as it might require an additional step of raising the root to a certain power.

In the given example \(8^{\frac{1}{3}}\), simplification requires you to find the cube root of 8. Since this is a perfect cube (\(2^3 = 8\)), it simplifies to 2. For other expressions, such as those with larger numerators, you may need to find the root first and then raise that result to the power of the numerator. Always remember to express the final result in its simplest form, ensuring no radical sign remains if the root can be found precisely and there are no unnecessary fractional powers in your answer.