Problem 34 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. $$ (8 \cdot 27)^{1 / 3} $$

Step-by-Step Solution

Answer

The given expression evaluates to 6.

Step 1: Multiply inside the parentheses

First, calculate the multiplication of 8 and 27. The answer is 216.

Step 2: Calculate the cube root

Then, calculate the cube root of 216. In mathematics, the cube root of a number x is a number y such that y^3 = x. So, we need to find y for 216. The cube root of 216 is 6 as 6^3 = 216.

Key Concepts

Cube Root

The cube root is a mathematical operation that, when applied to a number, gives another number which when multiplied by itself three times yields the original number. For example, if you're looking for the cube root of 216, the result should be a number, say y, such that \(y \cdot y \cdot y = 216\). In this case, 6 is the cube root because \(6^3 = 6 \cdot 6 \cdot 6 = 216\).

To find the cube root of any number, consider the following steps:

• Identify a number that, when raised to the power of 3, gives the original number.
• Check by multiplying the potential cube root by itself three times to ensure it results in the original number.
• If manually calculating a cube root seems difficult, you can use a calculator that supports this operation.

Cube roots are denoted using the radical symbol with a small three (\(\sqrt[3]{x}\)). Understanding cube roots is crucial as it extends into various branches of mathematics, including geometry and algebra.

Multiplication

Multiplying numbers is one of the basic arithmetic operations and can be thought of as repeated addition. In the exercise given, you multiply 8 by 27 to get 216.

Here's a simple walkthrough to solidify your understanding of multiplication:

• Start with two numbers that you need to multiply. In this case, 8 and 27.
• Think of one number as the number of groups and the other as the size of each group. For instance, assume 8 groups of 27.
• Use the multiplication table if you're learning or try breaking numbers into smaller, more manageable parts to multiply and then add the results.
• For mental math, break down larger numbers. For instance, think of 27 as 20 + 7, multiply each part by 8 separately, and then add the results: \(8 \times 20 = 160\) and \(8 \times 7 = 56\). So, \(160 + 56 = 216\).

Mastering multiplication helps in solving more complex mathematical problems and is essential in everyday life situations, like calculating prices or reading measurements.

Arithmetic Operations

Arithmetic operations form the foundation of mathematics, with multiplication and cube roots being essential operations you often encounter. Arithmetic operations include addition, subtraction, multiplication, and division. Each operation has its own rules and uses:

• Addition: Combining two or more numbers to get a sum.
• Subtraction: Taking away one number from another to find the difference.
• Multiplication: A quick way of adding the same number several times.
• Division: Splitting a number into equal parts.

Understanding how these basic operations work together is key to performing tasks like evaluating expressions. The order in which these operations occur is known as the order of operations. It's remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right)).

Developing a firm grasp on arithmetic operations enables deeper exploration into more advanced topics such as algebra, calculus, and beyond. Practicing these operations helps in building mathematical aptitude and problem-solving skills.