Problem 26 - Algebra A Combined Function

Question

Find each cube root. $$ \sqrt[3]{27} $$

Step-by-Step Solution

Answer

The cube root of 27 is 3.

Step 1: Understand the Problem

We need to find the cube root of 27, which is represented as \( \sqrt[3]{27} \). This means finding a number \( x \) such that \( x^3 = 27 \).

Step 2: Recall Cube Numbers

Recall the cubes of some smaller integers. Specifically, \( 2^3 = 8 \) and \( 3^3 = 27 \). From this, we can see that 3 cubed equals 27.

Step 3: Solve for the Cube Root

From Step 2, we identified that \( 3^3 = 27 \), meaning the cube root of 27 is 3. Therefore, \( \sqrt[3]{27} = 3 \).

Step 4: Verify the Solution

Verify your answer by cubing the solution: \( 3^3 = 27 \). Since this statement is true, our solution is correct.

Key Concepts

Exponentiation

Exponentiation is a mathematical operation, involving a base and an exponent. It is written as \( a^n \), where \( a \) is the base and \( n \) is the exponent. This simply means multiplying \( a \) by itself \( n \) times. For example, \( 3^2 \) is 3 multiplied by itself once, resulting in 9.
Understanding how to work with exponents is essential in solving expressions, simplifying problems, and finding roots.

• When the exponent is 1, the result is the base itself (e.g., \( a^1 = a \)).
• When the exponent is 0, the result is always 1, if the base is not zero (e.g., \( a^0 = 1 \)).
• If \( n \) is positive, you multiply the base by itself to get the result.

Mastering the basic principles of exponentiation helps in various mathematical areas, including algebra, calculus, and beyond.

Integer Cubes

Integer cubes are the results of raising integer numbers to the power of three. Essentially, when you cube a number, such as \( n \), you perform the operation \( n^3 = n \times n \times n \). For instance, cubing 2 gives you \( 2^3 = 8 \), and cubing 3 gives you \( 3^3 = 27 \).
Cubed numbers are a vital aspect of number theory and help in understanding the properties of numbers more deeply.

• \( 1^3 = 1 \)
• \( 2^3 = 8 \)
• \( 3^3 = 27 \)

These cubes of small integers are particularly important as benchmarks for identifying cube roots quickly.

Three-Dimensional Roots

The concept of three-dimensional roots, or cube roots, involves finding a number which, when cubed, gives the original number. The notation for cube roots is \( \sqrt[3]{x} \). For example, when asked to find \( \sqrt[3]{27} \), you are essentially searching for a number which satisfies \( x^3 = 27 \). In this case, the cube root of 27 is 3 because \( 3^3 = 27 \).
Finding cube roots is closely related to solving equations and inequalities in algebra. It involves understanding and applying inverse relationships, as cube roots are the inverse operation of cubing a number.

• Practical uses include measurements in geometry, where you deal with cubes and volumes.
• Cube roots help in simplifying algebraic expressions.

Mastering cube roots allows for more flexibility and understanding in diverse mathematical contexts.

Basic Algebra Concepts

Algebra is a branch of mathematics that uses symbols and letters to represent numbers and quantities in formulae and equations. In the context of finding roots like the cube root, we leverage several fundamental algebra concepts, such as:

• Variables: Symbols that are used to represent numbers.
• Equations: Mathematical statements that show the equality between two expressions.
• Operations: The arithmetic processes such as addition, subtraction, multiplication, and division.

For example, when working with the equation \( x^3 = 27 \), we are identifying the variable \( x \) that satisfies the equation. It's a straightforward application of these basic concepts that showcases the practical use of algebra in solving real-world problems.
Algebra builds the foundation for higher mathematics, including calculus, and is a powerful tool for problem-solving.