Problem 26 - Intermediate Algebra

Question

Find each cube root. $$ \sqrt[3]{x^{15}} $$

Step-by-Step Solution

Answer

The cube root of \( x^{15} \) is \( x^5 \).

Step 1: Understand the Problem

We're given the expression \( \sqrt[3]{x^{15}} \), which means we need to find the cube root of \( x^{15} \). This involves finding a value that, when raised to the power of 3, results in \( x^{15} \).

Step 2: Use the Cube Root Property

The property of cube roots allows us to simplify the expression: \( \sqrt[3]{x^n} = x^{n/3} \). Applying this to our problem, we have \( \sqrt[3]{x^{15}} = x^{15/3} \).

Step 3: Simplify the Exponent

Now we need to simplify the exponent \( 15/3 \). Performing the division gives us \( 5 \). Therefore, \( x^{15/3} = x^5 \).

Step 4: Write the Final Answer

Once simplified, the cube root of \( x^{15} \) is \( x^5 \).

Key Concepts

Understanding Exponents

Exponents are a fundamental concept in algebra and mathematics in general. They are used to describe how many times a number, known as the base, is multiplied by itself. For example, in \(x^3\), the base \(x\) is multiplied by itself three times: \(x \times x \times x\). Exponents make expressions shorter and calculations easier to follow.

• Base and Exponent: The number being multiplied is the base, and the exponent tells us how many times to multiply the base by itself.
• Properties of Exponents: A key property is \(x^{a+b} = x^a \cdot x^b\), which allows us to add exponents when multiplying similar bases.
• Examples: If you have \(2^3 = 2 \times 2 \times 2 = 8\), this takes advantage of the exponent to abbreviate the multiplication process.

Understand these basics, and calculations involving powers will become much simpler.
When you encounter problems involving cubes or cube roots, the principle remains the same but in reverse.

Simplification Techniques

Simplification in mathematics involves reducing an expression to its simplest form, making it easier to understand and work with. In the context of cube roots, simplification might involve using properties of exponents for ease.

• Cube Root Simplification: The rule \(\sqrt[3]{x^n} = x^{n/3}\) is a powerful tool. If you apply this to \(\sqrt[3]{x^{15}}\), you simplify directly to \(x^5\), as you divide the exponent 15 by 3.
• Step-by-Step: Reduce step by step. Even if the outcome seems obvious, knowing each step ensures accuracy, especially in complex expressions.
• Consistency: Maintain precision by consistently applying rules of exponents across all instances.

Appropriate simplification helps in making expressions manageable, which is crucial for both basic and advanced algebra.

Algebraic Expressions Usage

Algebraic expressions are combinations of numbers, variables, and operations. They give us a way to represent relationships and solve for unknowns. Expressions like \(x^{15}\) are common in algebra, requiring manipulation via rules like those of exponents or simplification to manage.

• Variables and Constants: Letters such as \(x\) are used as variables, representing unknowns or values that can change. Constants like \(15\) stay the same.
• Manipulating Expressions: Being able to rewrite expressions using exponents or roots, as seen in simplifying \(\sqrt[3]{x^{15}} = x^5\), is a basic yet essential skill.
• Practical Applications: Algebraic expressions are used to model real-world situations. For example, expressions can represent geometric formulas or growth patterns.

Understanding how to handle algebraic expressions effectively opens doors to solving equations and understanding functional relationships.