Problem 35 - Algebra 2

Question

Simplify. $$ \sqrt[3]{8 a^{3} b^{3}} $$

Step-by-Step Solution

Answer

The simplified form is \(2ab\).

Step 1: Recognize Perfect Cubes

The expression under the cube root is \(8a^3b^3\). Notice that \(8 = 2^3\), \(a^3\) is a perfect cube, and \(b^3\) is also a perfect cube. Identifying these perfect cubes helps us simplify the expression.

Step 2: Apply Cube Root Property

The cube root of \(x^3\) is \(x\). Therefore, \(\sqrt[3]{2^3} = 2\), \(\sqrt[3]{a^3} = a\), and \(\sqrt[3]{b^3} = b\).

Step 3: Simplify the Expression

Substitute the simplified components from Step 2 back into the expression: \(\sqrt[3]{8a^3b^3} = \sqrt[3]{2^3} \times \sqrt[3]{a^3} \times \sqrt[3]{b^3} = 2 \times a \times b\).

Step 4: Final Result

Combine the simplified components to get the final result: \(2ab\). This means the expression \(\sqrt[3]{8a^3b^3}\) simplifies to \(2ab\).

Key Concepts

Perfect Cubes

In the world of mathematics, a perfect cube is a number that can be expressed as the cube of an integer. This means if you multiply an integer by itself three times, you get a perfect cube. For instance, let's consider the number 8. It can be represented as \( 2^3 \), which indicates that 8 is a perfect cube because it's the result of multiplying 2 by itself thrice.

Perfect cubes play a significant role in simplifying expressions, especially when dealing with cube roots. Recognizing perfect cubes allows us to easily apply the cube root, simplifying complex expressions into more manageable forms.

• Example: \( 27 = 3^3 \)
• Example: \( 64 = 4^3 \)
• Example: \( 125 = 5^3 \)

In algebra, perfect cubes often appear in equations, making it essential to identify and manipulate them for simplification processes. Just like \( 8a^3b^3 \) in our exercise, each component needs to be perfect cubes to ease simplification.

Cube Root

The cube root of a number is a value that, when multiplied by itself twice, gives the original number. Understanding cube roots is fundamental when you want to simplify radicals. In algebra, the cube root is denoted as \( \sqrt[3]{x} \).For instance, finding the cube root of a perfect cube such as 8 is straightforward. Since 8 can be written as \( 2^3 \), its cube root is 2. This principle applies to variables as well.

• \( \sqrt[3]{2^3} = 2 \)
• \( \sqrt[3]{a^3} = a \)
• \( \sqrt[3]{b^3} = b \)

The property of cube roots is used to deconstruct expressions like \( \sqrt[3]{8a^3b^3} \). By taking the cube root of each perfect cube individually, you simplify the expression into products of their cube roots, showing how cube roots simplify radical expressions substantially.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations such as addition, multiplication, and exponentiation. In the context of simplifying radicals, algebraic expressions are transformed using properties of exponents and roots.In the expression \( \sqrt[3]{8a^3b^3} \), each component within the cube root represents an algebraic term or factor. Simplification requires applying mathematical properties systematically:

• Identify perfect cubes.
• Use the cube root to extract each term.
• Simplify the result into a straightforward expression.

For the expression given, recognizing that 8, \( a^3 \), and \( b^3 \) are perfect cubes permits us to write: * \( 8 = 2^3 \) * \( a^3 = a^3 \) * \( b^3 = b^3 \)This insight facilitates the straightforward simplification way to achieve the result \( 2ab \), illustrating how strategies such as identifying factors can unify and simplify algebraic tasks.