Problem 18 - Algebra 1: Concepts and Skills

Question

Evaluate the expression without using a calculator. $$ \sqrt[3]{8} $$

Step-by-Step Solution

Answer

The cubic root of 8 is 2.

Step 1: Understanding the operation

A cubic root, represented as \( \sqrt[3]{x} \), is the number that, when multiplied by itself twice, gives the number \(x\). In this exercise, the goal is to find the cubic root of 8, which means finding a number that, when multiplied by itself twice, results in 8.

Step 2: Evaluating the cubic root

We know the cubic numbers up to 10: 1 (which is \(1^3\)), 8 (which is \(2^3\)) and 27 (which is \(3^3\)). Here, it can be seen that 8 is a cubic number, and it is the result of 2*2*2. Hence, 2 is the cubic root of 8.

Key Concepts

Evaluating Expressions

When you evaluate an expression, you're figuring out what that expression means or equals. Here, we are dealing with the cubic root of 8: \( \sqrt[3]{8} \). Evaluating this expression means finding a number that, when multiplied by itself twice, gives you 8. This process requires simplification without a calculator, allowing you to sharpen your mental math skills.

• Look for familiar numbers: It's helpful to think about numbers that, when cubed, equal 8.
• Use logical reasoning: Remembering small set of cubic numbers can simplify the evaluation task.

Evaluation is like solving a puzzle. You piece together clues (cubic numbers in this case) to find your answer. The goal here is sharpening your evaluation skills, which is useful for breaking down more complex math problems.

Cubic Numbers

Cubic numbers are numbers that can be expressed as \( x^3 \), meaning a number \( x \) multiplied by itself twice. For example:

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

These numbers quickly escalate in size, making it useful to remember the smaller ones by heart. Understanding cubic numbers helps you recognize when they appear in expressions and can simplify problems quickly. Here, knowing that 8 is the same as \(2^3\) allows us to identify that 2 is the cubic root of 8.

Exponents

Exponents are a way of showing repeated multiplication of the same number. For instance, \( x^3 \) means \(x\) is multiplied by itself two more times, i.e., \( x \times x \times x \). This is fundamental in understanding cubic roots.

• Exponents simplify large multiplications.
• They provide a way to express large numbers succinctly.

When dealing with exponents, it helps to recall and understand smaller exponents like squares and cubes. This not only aids calculation but also enhances your grasp of number properties. Recognizing that \( 2^3 = 8 \) helps quickly identify that the cubic root of 8 is 2, making exponent skills crucial for efficient problem solving.