Problem 55 - Algebra A Combined Function

Question

Simplify. Assume that the variables represent any real number. $$ \sqrt[3]{x^{3}} $$

Step-by-Step Solution

Answer

The expression \( \sqrt[3]{x^3} \) simplifies to \( x \).

Step 1: Understand the Problem

We need to simplify the expression \( \sqrt[3]{x^3} \). This means we are looking for the cube root of \( x^3 \). The cube root of a number is a value that, when cubed, gives the original number.

Step 2: Apply the Cube Root Property

Recall that the cube root of \( a^3 \) is \( a \). Therefore, \( \sqrt[3]{x^3} \) simplifies to \( x \). This property works for real numbers, so it is valid here since the problem assumes \( x \) represents any real number.

Step 3: Write the Simplified Expression

Using the cube root property from Step 2, rewrite the original expression \( \sqrt[3]{x^3} \) as \( x \). Thus, the simplified form of the expression is \( x \).

Key Concepts

Real Numbers

Real numbers are an essential part of mathematics and are used to represent continuous quantities. They include both rational and irrational numbers:

• Rational numbers can be expressed as fractions, such as \( \frac{3}{4} \) or \( -2 \).
• Irrational numbers cannot be written as simple fractions, examples include \( \sqrt{2} \) and \( \pi \).

Real numbers can be positive, negative, or zero, and they fill the number line without gaps. When working with expressions like \( \sqrt[3]{x^3} \), it's important to remember that even though \( x \) can be any real number, it must be a real number to validate the cube root property. By understanding the scope of real numbers, you can simplify expressions with confidence.

Cube Root Property

The cube root property is a fundamental concept in algebra. It states that the cube root of a number raised to the power of three (\( a^3 \)) is simply the number itself (\( a \)).Consider this example: If you have \( a = 2 \), then:

• \( a^3 = 2^3 = 8 \)
• The cube root of \( 8 \) is \( \sqrt[3]{8} = 2 \)

The cube root property is especially useful when simplifying expressions. For instance, when you have \( \sqrt[3]{x^3} \), you can straightforwardly write it as \( x \) because the cube root "undoes" the cubing of \( x \). This makes working with expressions and equations more manageable, especially when dealing with complex algebraic problems.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form. This process makes expressions easier to understand and solve. Let's break down how you can simplify cube root expressions like \( \sqrt[3]{x^3} \):1. **Identify the operation**: Recognize you're dealing with a cube root, which asks for a number that, when cubed, produces the original quantity.2. **Apply properties**: Use the cube root property to simplify. Since \( \sqrt[3]{x^3} \) equals \( x \), you leverage the property that \( \sqrt[3]{a^3} = a \).When simplifying:

• Focus on recognizing the patterns (like \( x^3 \)) that are already in a form that relates directly to root operations.
• Remember that assumptions like having real numbers help validate these properties and ensure accurate simplification.

These steps help you systematically simplify expressions, making complex mathematical problems easier to navigate.