Problem 8 - Algebra 1: Concepts and Skills

Question

Evaluate the expression without using a calculator. $$ \sqrt[3]{343} $$

Step-by-Step Solution

Answer

The cube root of 343 is 7.

Step 1: Recognize the number as a cube

The first step is to determine if 343 is a cubic number. Cubic numbers are those numbers which can be written as the cube of a number. The cubes of the numbers 1 through 10 are as follows: \[1^3 = 1, 2^3 = 8, 3^3 = 27, 4^3 = 64, 5^3 = 125, 6^3 = 216, 7^3 = 343, 8^3 = 512, 9^3 = 729, 10^3 = 1000\] Looking at these values, it can be seen that \(7^3 = 343\).

Step 2: Identify the cube root

The cube root of a number is the value that, when cubed (multiplied by itself twice), will yield the original number. Since 343 is the cube of 7. Therefore, the cube root of 343 is 7, written mathematically as \(\sqrt[3]{343} = 7\).

Key Concepts

Cube Root

A cube root of a number is the special value that when multiplied by itself three times equals the original number. In mathematical terms, finding the cube root of a number is the reverse process of cubing a number.
For example, consider the number 343. We observed in our problem that 343 can be written as \( 7^3 \). Thus, 7 is the cube root of 343. This is expressed mathematically as \( \sqrt[3]{343} = 7 \).
Cube roots can be found for both positive and negative numbers. If you take the cube root of -8, for instance, you'll find that it is -2 because \( (-2)^3 = -8 \).
Understanding cube roots is essential in mathematics, especially in solving equations involving volume, geometry, or physics, where cubic measurements are essential.

Mathematical Evaluation

Mathematical evaluation involves the process of calculating the value of an expression. This can often include identifying relationships between numbers or expressions, such as factoring or recognizing perfect cubes.
In the given problem, you are tasked with finding the cube root of 343 without a calculator. This requires identifying 343 as a cubic number \( 7^3 \), meaning the number 7, when cubed, results in 343.
To evaluate this expression without digital aid, it helps to memorize common cubes of small numbers. These include values like \( 2^3 = 8 \), \( 3^3 = 27 \), \( 4^3 = 64 \), etc.
Evaluating mathematical expressions manually not only sharpens mental arithmetic but also deepens your understanding of number properties and relationships.

Exponentiation

Exponentiation is the mathematical operation involving two numbers, the base and the exponent. It's a way of representing repeated multiplication of a number by itself.
For instance, in \( 7^3 \), 7 is the base, and 3 is the exponent, which means 7 is multiplied by itself two more times (\( 7 \times 7 \times 7 \), resulting in 343).
Exponentiation is key to understanding the concept of powers and roots, including square roots and cube roots. When the number is raised to the power of 3, it becomes a cube, hence the process of finding a cube root is essentially reversing this exponentiation process.
Grasping exponentiation helps in various mathematical operations as it's frequently used in algebra, calculus, and even scientific computations involving growth and decay models.