Problem 79 - College Algebra

Question

In Exercises \(77-84,\) evaluate each expression without using a calculator. $$8^{1 / 3}$$

Step-by-Step Solution

Answer

The cube root of 8 is \(2\). Therefore, \(8^{1 / 3} = 2\).

Step 1: Understand the Exponent

The expression given is \(8^{1 / 3}\). The exponent of 1/3 corresponds to the cube root in mathematics. This means we are looking for a number that, when multiplied by itself three times, will equal 8.

Step 2: Solve for the Cube Root

There are few basic cube roots that you usually learn by heart, one of them being \(2^{3} = 8\), which means that the cube root of 8 is 2. Therefore, \(8^{1 / 3} = 2\).

Key Concepts

Exponentiation

Exponentiation is a mathematical operation that involves raising a number, known as the base, to a power, which is the exponent. It is denoted by the base followed by a superscript exponent. For example, in the expression \(2^3\), 2 is the base and 3 is the exponent, indicating that 2 should be multiplied by itself two additional times (2 x 2 x 2).

When an exponent is a fraction, such as \(1/3\), it represents a root. Specifically, \(8^{1/3}\) means the cube root of 8. Exponentiation is crucial in various mathematical fields and its properties are widely used to simplify and solve equations. Understanding how to manipulate exponents—including fractional exponents—is fundamental for higher-level math and various applications in science and engineering.

Cube Root

The cube root of a number is a value that, when multiplied by itself three times, gives the original number. It is the inverse operation of cubing a number. For instance, if \(2^3 = 8\), then the cube root of 8 is 2, written as \(\sqrt[3]{8} = 2\).

Cube roots can be thought of as a special case of exponentiation where the exponent is \(1/3\). It's important to recognize that every real number has exactly one real cube root, which includes negative numbers. For example, the cube root of -27 is -3 because \( (-3)^3 = -27\). Recognizing basic cube roots of small perfect cubes like 1, 8, 27, etc., can be very useful when evaluating expressions without a calculator.

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, operations, and sometimes grouping symbols that show operations to be performed to produce a value. Expressions can be simple, such as \(2 + 3\), or more complex, involving exponents and roots like \(8^{1/3}\).

To evaluate an expression means to find its value. This requires knowledge of the order of operations, also known as PEMDAS (Parentheses, Exponents, Multiplication and Division from left to right, Addition and Subtraction from left to right). Evaluating expressions without a calculator can improve mental math skills and a deeper understanding of fundamental math concepts. Practice with frequent operations like exponentiation, cube roots, and perform operations mentally or on paper is recommended for mastery.