Problem 80 - College Algebra

Question

In Exercises \(77-84,\) evaluate each expression without using a calculator. $$27^{1 / 3}$$

Step-by-Step Solution

Answer

The cube root of 27, identified by the expression \(27^{1 / 3}\), is 3.

Step 1: Identify the task

The given expression is \(27^{1 / 3}\), so we want to find the cube root of 27.

Step 2: Find a number whose cube is 27

In this case, that number is 3, because \(3^3 = 27\)

Step 3: Check the solution

Substitute 3 to \(x^{1 / 3}\) in place of 27: \(3^{1 / 3} = 3\). Therefore, \(27^{1 / 3} = 3\)

Key Concepts

Exponentiation

Exponentiation is a mathematical operation involving two numbers, the base and the exponent or power. The exponent signifies how many times the base is multiplied by itself. For example, in the expression \(2^3\), 2 is the base and 3 is the exponent, meaning that 2 is multiplied by itself 3 times: \(2 \times 2 \times 2 = 8\).

In our exercise, we have the expression \(27^{1/3}\), which denotes the cube root of 27. This can be interpreted as finding the number that, when raised to the power of 3, equals 27. The answer, as revealed through the calculation in the provided steps, is 3. Remember, any number to the power of \(1/n\) is the nth root of the number.

Radicals

Radicals involve the use of a root symbol to indicate the root of a number. In algebra, the most common radicals are square roots and cube roots, represented by symbols \(\sqrt{}\) and \(\sqrt[3]{}\), respectively.

The expression \(27^{1/3}\) is equivalent to the radical \(\sqrt[3]{27}\). Solving a radical means determining the non-negative root of a number. The cube root of a number is a special case where we're looking for a number that multiplied by itself three times gives the original number. As established, the cube root of 27 is 3, because \(3 \times 3 \times 3 = 27\). When dealing with radicals, particularly in algebra, it's important to consider both the index (which in the case of a cube root is 3) and the radicand (the number under the root, which here is 27).

Algebraic Expressions

Algebraic expressions are combinations of constants, variables, and operations like addition, subtraction, multiplication, division, exponentiation, and radicals. They are fundamental in representing mathematical concepts and solving equations.

In the context of the exercise, \(27^{1 / 3}\) is an algebraic expression with a base of 27 and an exponent of \(1/3\), which indicates the cube root. Algebraic expressions can be simplified or evaluated, and understanding how to manipulate these expressions is essential in algebra. For example, we solved the cube root of 27 by evaluating the algebraic expression \(27^{1 / 3}\), which simplified to 3. This is a direct application of understanding both exponentiation and radicals as they apply to algebraic expressions.