Problem 5 - Intermediate Algebra

Question

Evaluate each of the following. For example, \(\sqrt{25}=5\). \(\sqrt[3]{27}\)

Step-by-Step Solution

Answer

The cube root of 27 is 3.

Step 1: Identify the Expression

The given expression is the cube root of 27, which is written as \( \sqrt[3]{27} \). This expression asks for the number which, when multiplied by itself three times, equals 27.

Step 2: Recognize Common Cube Values

Recall common cube values to quickly identify potential solutions. Some common cubes include \( 1^3 = 1 \), \( 2^3 = 8 \), and \( 3^3 = 27 \). This hints that the cube root of 27 might be 3.

Step 3: Verify the Cube Root

To verify, calculate \( 3^3 \,= 3 \times 3 \times 3 \,= 27 \). Since the answer equals the original 27, we confirm that our potential solution is correct.

Step 4: Conclude the Result

Since multiplying 3 by itself three times gives 27, we have confirmed that the cube root of 27 is 3.

Key Concepts

Cube Root Calculation

A cube root is a special number that, when used three times in a multiplication, gives the original number. In simpler terms, if you think about breaking down a cube packed tightly with smaller cubes, finding the cube root means figuring out how many small cubes fit along one side.
To calculate cube roots, look for a number that, when multiplied by itself twice more, equals the number in question. This process is crucial for evaluating the cube root of expressions, where you often don't have a straightforward visible solution.
For instance, to find the cube root of 27, you're essentially asking: "What number do I multiply by itself, and again, that results in 27?" In such cases, recognizing common cube values—numbers like 1, 8, 27, 64, etc.—helps us find a quick solution.

Common Cube Values

Identifying common cube values is a vital piece of knowledge when working with cube roots. These are numbers that arise from cubing whole numbers. By remembering a few key cubes, you can quickly solve cube root problems without complex calculations.
Some commonly known cubes are:

• 1 cubed is \(1^3 = 1\)
• 2 cubed is \(2^3 = 8\)
• 3 cubed is \(3^3 = 27\)
• 4 cubed is \(4^3 = 64\)
• 5 cubed is \(5^3 = 125\)

Having these values at your fingertips makes solving expressions involving cube roots much faster. Remember, when you get a number, check if it's a cube of a familiar integer to quickly determine its cube root.

Evaluating Expressions

Evaluating expressions that involve cube roots involves recognizing patterns and applying basic multiplication principles. Often, these problems require you to estimate or factor numbers to identify an appropriate cube root.
The goal is to simplify the number inside the radical sign (\( \sqrt[3]{...}\)) until you find a perfect cube. Upon finding such a number, the cube root can be directly simplified to the base that was cubed to get it.
For example, with the expression \( \sqrt[3]{27} \), we identify 27 as a common cube, specifically, the cube of 3. Thus the expression simplifies to 3, which becomes the final answer. This process involves recalling common cubes and confirming by cubing the potential answers to ensure correctness.