Problem 1 - Algebra 2

Question

Simplify. $$ \sqrt[3]{64} $$

Step-by-Step Solution

Answer

The simplified form of \(\sqrt[3]{64}\) is 4.

Step 1: Understand the Problem

We need to find the cube root of 64. This means we are looking for a number that, when multiplied by itself three times, equals 64.

Step 2: Identify Perfect Cube Candidates

Let's consider a few numbers and their cubes: - \(2^3 = 8\) - \(3^3 = 27\)- \(4^3 = 64\)These calculations suggest that 4 is a good candidate because \(4^3 = 64\).

Step 3: Verify the Cube

We suspect that \(4\) is the cube root of \(64\). To confirm, we calculate \(4 imes 4 imes 4 = 16 imes 4 = 64\).

Step 4: Conclude the Simplification

Since \(4^3 = 64\), we have verified that \(4\) is indeed the cube root of \(64\). Thus, the simplified form of \(\sqrt[3]{64}\) is \(4\).

Key Concepts

Simplifying Radicals

When it comes to simplifying radicals, particularly cube roots, the goal is to find a simpler or more manageable form of a radical expression. A radix, often known as a root, helps determine what number, when multiplied a certain number of times, will recreate the original number. For cube roots, the objective is to find a number that, when multiplied by itself three times, equals the number under the cube root (radicand).

Here's how you can simplify cube roots effectively:

• First, identify the radicand, which is the number inside the radical; in our example, it is 64.
• Next, find a number that, when multiplied by itself three times, will give the radicand back. This number simplifies the radical.
• If you successfully find such a number, you've simplified the cube root.

Finding the cube root of 64 involves recognizing that 4 multiplied by itself three times results in 64, thus simplifying \( \sqrt[3]{64} \) into just 4.

Perfect Cubes

Understanding perfect cubes is crucial when dealing with cube roots. A perfect cube is a number that can be expressed as an integer raised to the power of three. Identifying these numbers is essential when simplifying cube roots because they make computation straightforward.

Some characteristics of perfect cubes include:

• They can always be written as \( n^3 \) for some integer \( n \).
• They have relatively simple factorizations in terms of prime multipliers.
• The cube roots of perfect cubes are always whole numbers.

In the case of our example, 64 is a perfect cube because it is equal to \( 4^3 \). Recognizing that 64 is a perfect cube directly tells you that 4 is its cube root without needing complex calculations. Hence, knowing your perfect cubes well can significantly ease the task of finding cube roots.

Algebraic Expressions

Algebraic expressions often involve operations like finding cube roots. Understanding these expressions is essential to simplify them effectively, especially those involving radicals.

When simplifying an algebraic expression involving a cube root:

• Identify any perfect cubes within the expression to simplify the radical components quickly.
• Use rules of exponents to consolidate any remaining terms.
• Remember to always simplify expressions by combining like terms and maintaining the hierarchy of operations.

Incorporating cube roots into algebraic expression requires navigating variables and constants with careful attention to simplification rules. By knowing how to identify and work with perfect cubes, simplifying these expressions becomes a straightforward task, allowing for easier manipulation and interpretation of the algebraic form.