Problem 42 - Intermediate Algebra

Question

Simplify. See Examples 3 and 4 $$ \sqrt[3]{64 y^{9}} $$

Step-by-Step Solution

Answer

The simplified expression is \(4y^3\).

Step 1: Identify the Cube Root

The problem involves finding the cube root of the expression \(64y^9\). Write this as \( (64y^9)^{1/3} \) to signify raising the expression to the power of \(1/3\).

Step 2: Simplify the Radical Expression

Recognize that \(64\) is a perfect cube, as \(64 = 4^3\). Also, express the power of \(y\) under the cube root: \(y^9 = (y^3)^3\). Thus, the expression inside the cube root can be rewritten as \((4^3 (y^3)^3)^{1/3}\).

Step 3: Apply Cube Root to Each Component

Apply the cube root to \(4^3\) and \((y^3)^3\) separately: \( (4^3)^{1/3} = 4 \) and \(((y^3)^3)^{1/3} = y^3\).

Step 4: Combine the Simplified Results

Combine the results from the previous step to obtain the fully simplified expression: \(4y^3\). This is your simplified cube root expression.

Key Concepts

Simplifying Radical Expressions

Radical expressions may look complicated, but by understanding the basics, you can simplify them with ease. Simplifying radical expressions involves breaking them down to their most basic form. The goal is to rewrite the expression so there's no longer a radical or to reduce the degree of the radical.

• In our original exercise, the cube root is represented as \( \sqrt[3]{64y^9} \).
• To simplify, we rewrote this as \( (64y^9)^{1/3} \), which highlights that we are dealing with a radical expression raised to the power of \(1/3\).

• This changed the complex radical expression into something we can manage with exponents.

Understanding this approach can help you tackle more complex radical expressions with confidence.

Perfect Cubes

Perfect cubes are numbers or expressions that can be expressed as some base number raised to the power of three. Recognizing these helps in simplifying cube roots or cube powers.

• For example, the number \(64\) is a perfect cube, since \(64 = 4^3 \).
• This means \(64\) can be conveniently represented in a form that simplifies the process of finding cube roots.
• Additionally, expressions involving variables can also be perfect cubes, such as \(y^9 = (y^3)^3 \).

These forms make it easy to simplify radical expressions involving cube roots.

Exponent Rules

Exponent rules are fundamental mathematical properties that help in simplifying expressions involving powers. Mastery of these rules is critical for dealing with algebraic expressions and simplifying radical expressions.
One key exponent rule is the power of a power rule:

• This rule states that \((a^m)^n = a^{mn} \), where \(a\) is the base, and \(m\) and \(n\) are exponents.
• Our solution utilized this when simplifying \((y^3)^3 \).

Applying the cube root \(((y^3)^3) = y^9\) demonstrates effectively using these rules. This exponent rule lets us simplify complex powers into simpler forms with ease.

Algebraic Expressions

An algebraic expression is a mathematical phrase that includes numbers, variables, and operations. Simplifying algebraic expressions involves combining like terms and reducing expressions to their simplest form.
In our example:

• The original expression \(64y^9\) includes numbers and a variable raised to a power.
• The simplification process involves evaluating this expression step-by-step while applying exponent rules and recognizing perfect cubes.

This process yielded the final simplified result, \(4y^3\), which is a clean and simplified algebraic expression. Understanding how to manage algebraic expressions is foundational in algebra and assists with solving a variety of math problems.