Problem 42 - Algebra A Combined Function

Question

Use the quotient rule to simplify. Assume that all variables represent positive real numbers. $$ \sqrt[3]{128 y^{10}} $$

Step-by-Step Solution

Answer

The simplified expression is \( 2^{7/3} y^{10/3} \).

Step 1: Write in Exponent Form

The expression \( \sqrt[3]{128 y^{10}} \) can be rewritten in exponential form. Recall that \( \sqrt[n]{a} = a^{1/n} \). Thus, the expression becomes \( (128 y^{10})^{1/3} \).

Step 2: Apply the Power of a Product Property

The power of a product states that \( (ab)^n = a^n b^n \). Apply this to \( (128 y^{10})^{1/3} \) to get \( 128^{1/3} \times (y^{10})^{1/3} \).

Step 3: Simplify Each Part Separately

First, find \( 128^{1/3} \). Since \( 128 = 2^7 \), we have \( 128^{1/3} = (2^7)^{1/3} = 2^{7/3} \).Now simplify \( (y^{10})^{1/3} \), which equals \( y^{10/3} \).

Step 4: Combine the Simplified Parts

Combine the separate simplified results from Step 3: Therefore, the simplified expression is \( 2^{7/3} \, y^{10/3} \). This can also be presented as \( 2^{2 + 1/3} \, y^{3 + 1/3} \) or \( 4 \sqrt[3]{2} \times y^3 \sqrt[3]{y} \) if decomposed further.

Key Concepts

Exponent Form

When dealing with roots such as cube roots or square roots, we can rewrite them using exponents to make them easier to manage, especially when simplifying expressions. For example, the cube root of a number can be written as an exponent with a fractional power. This is generally expressed with the formula \( \sqrt[n]{a} = a^{1/n} \). By rewriting roots in this manner, we allow ourselves to use exponential rules like the Power of a Product or Quotient Rule more effectively.

So, in our example, \( \sqrt[3]{128 y^{10}} \) becomes \( (128 y^{10})^{1/3} \). This conversion is the first crucial step in simplifying the expression as it lays the groundwork for applying further exponent rules.

Power of a Product

The Power of a Product is an essential property that helps us distribute an exponent across a product of factors. It states that \( (ab)^n = a^n b^n \). In simpler terms, when a product is raised to a power, we can raise each factor to that power separately.

Applying this rule to our expression \( (128 y^{10})^{1/3} \) allows us to break it down into \( 128^{1/3} \times (y^{10})^{1/3} \). This separation into smaller parts makes it easier to handle computations or further simplifications with each component individually.

Using this property correctly helps in gradually simplifying expressions to reach a more manageable or desired format.

Simplifying Expressions

Simplifying expressions involves breaking down complex expressions into simpler or more succinct forms. For instance, after applying the Power of a Product, we next simplify each component separately.

• First, consider \( 128^{1/3} \). Since \( 128 = 2^7 \), \( 128^{1/3} \) becomes \( (2^7)^{1/3} = 2^{7/3} \).
• Next, tackle \( (y^{10})^{1/3} \) which simplifies to \( y^{10/3} \).

After simplifying these parts, we combine them to form the expression \( 2^{7/3} \, y^{10/3} \). If we want to express it differently, we can further adjust it to \( 2^{2 + 1/3} \, y^{3 + 1/3} \) which can also be broken down to \( 4 \sqrt[3]{2} \times y^3 \sqrt[3]{y} \), providing different perspectives on the simplified version.

Simplifying expressions often requires patience, attention to detail, and knowledge of properties like the Power of a Product to achieve a neat and concise result.