Problem 44 - Intermediate Algebra

Question

Simplify. See Examples 3 and 4 $$ \sqrt[5]{32 z^{12}} $$

Step-by-Step Solution

Answer

\( 2z^2 \sqrt[5]{z^2} \)

Step 1: Understand the Problem

We need to simplify the expression \( \sqrt[5]{32z^{12}} \). This means finding the fifth root of the product \( 32z^{12} \), which simplifies the expression under the radical sign.

Step 2: Simplify the Radicand

The expression inside the radical is \( 32z^{12} \). We can express 32 as \( 2^5 \) because 32 is \( 2 \times 2 \times 2 \times 2 \times 2 \). Thus, the expression becomes \( \sqrt[5]{2^5 z^{12}} \).

Step 3: Apply the Fifth Root

Since the fifth root of \( 2^5 \) is 2, we can simplify \( \sqrt[5]{2^5} \) to 2. This reduces the expression to \( 2 \cdot \sqrt[5]{z^{12}} \).

Step 4: Apply the Root to the Variable Part

For \( z^{12} \), we find the fifth root, which involves dividing the exponent by 5: \( (z^{12})^{1/5} = z^{12/5} \). To express this as an integer power and a remainder, we rewrite \( z^{12/5} \) as \( z^2 \cdot z^{2/5} \).

Step 5: Combine Results

The simplified expression is the product of the constants and remaining radical: \( 2z^2 \sqrt[5]{z^2} \). This is our simplified form of the original expression.

Key Concepts

Fifth Roots

In algebra, a "fifth root" refers to a number, which when multiplied by itself five times, returns the original number under the radical sign. For example, the fifth root of 32 is 2 because \(2^5 = 32\). Understanding roots is essential in simplifying expressions like \(\sqrt[5]{32z^{12}}\). To approach this concept, first identify numbers and variables under the radical sign.

• Determine if the number can be expressed as a power of another number, such as rewriting 32 as \(2^5\).
• Similarly, examine the variable part, like \(z^{12}\) in the expression, for simplification.

Expressing numbers and variables in exponential form helps in taking roots easily. Thus, the fifth root simplifies to 2, as we see when \(32 = 2^5\). This shows that fifth roots help simplify complex expressions by converting them into easier forms.

Exponent Rules

Understanding exponent rules is crucial for simplifying expressions such as \(\sqrt[5]{32z^{12}}\). Exponents provide a way to express repeated multiplication compactly. For instance, \(z^{12}\) signifies multiplying \(z\) by itself twelve times.
To simplify radical expressions involving exponents, it is essential to apply the property \[(x^a)^{1/b} = x^{a/b}\]This formula helps in rewriting radicals as fractional exponents, which then aids in simplification.
In our example, \((z^{12})^{1/5} = z^{12/5}\). Often, this fractional exponent can be split into an integer part and a remainder, such as rewriting \(z^{12/5}\) as \[z^2 \cdot z^{2/5}\]By understanding these rules, you can break down complex radical expressions into simpler parts.

Radicals in Algebra

Radicals play a significant role in algebra by simplifying expressions and solving equations that involve roots. The expression \(\sqrt[5]{32z^{12}}\) contains a radical, which represents the process of taking a root - in this case, the fifth root.When simplifying expressions with radicals, it’s helpful to:

• Express numbers and variables under the radical as a power of another number or variable, as seen with 32 and \(z^{12}\).
• Apply the correct root to both numerical and variable components.

Remember, taking the fifth root of a number is equivalent to raising it to the power of \(1/5\). This logic is applicable to the variable part as well. Thus, \(z^{12}\) inside a fifth root simplifies to \(z^{12/5}\), which can be broken down further.By learning how to manipulate radicals, you can ease complex algebraic expressions into more understandable forms, thus better grasping broader algebra concepts.