Problem 49 - Algebra A Combined Function

Question

Use the properties of exponents to simplify each expression. Write with positive exponents. $$ \left(4 u^{2}\right)^{3 / 2} $$

Step-by-Step Solution

Answer

The simplified expression is \(8u^3\).

Step 1: Apply Power of a Product Rule

The expression given is \( (4u^2)^{3/2} \). First, apply the power of a product rule, which states \((ab)^n = a^n b^n\). Thus, the expression becomes \( 4^{3/2} (u^2)^{3/2} \).

Step 2: Simplify the Numerical Base

Compute \( 4^{3/2} \). This can be broken down as \( (4^{1/2})^3 \). The square root of 4 is 2, so \( (4^{1/2}) = 2 \). Then, \( 2^3 = 8 \). Therefore, \( 4^{3/2} = 8 \).

Step 3: Simplify the Variable Exponent

Now simplify the variable part, \((u^2)^{3/2}\). Using the power of a power rule \((a^m)^n = a^{m\cdot n}\), we have \((u^2)^{3/2} = u^{2 \cdot (3/2)} = u^3\).

Step 4: Combine Simplified Parts

Combine the simplified parts from Steps 2 and 3. The expression becomes \( 8 \times u^3 \). Thus, the simplified expression is \( 8u^3 \).

Key Concepts

Power of a Product Rule

The power of a product rule is a fundamental concept in understanding exponents. It allows you to simplify expressions where an entire product is raised to an exponent. According to the rule,

• If you have an expression in the form of \((ab)^n\), this is equivalent to raising each factor inside the parentheses to the power separately: \(a^n \times b^n\).

This rule is particularly useful because it helps break down complex expressions into manageable parts.
Take, for instance, the expression \((4u^2)^{3/2}\). By applying the power of a product rule, you can separate this into two individual powers: \(4^{3/2}\) and \((u^2)^{3/2}\). This step is critical, as it simplifies the process of evaluating each component separately.
Breaking down complex expressions into simpler parts makes calculations easier and intuitive, enabling students to solve exponential problems efficiently.

Simplifying Exponents

Simplifying exponents involves reducing expressions to their simplest form using various properties of exponents. This process usually includes:

• Applying fundamental rules such as the power of a product and power of a power rules.
• Breaking down numerical bases and combining like terms when possible.

To simplify \(4^{3/2}\), consider first taking the square root of 4, which is 2, and then cubing the result.So, first compute \(4^{1/2} = 2\) followed by \(2^3 = 8\).
For the variable component \((u^2)^{3/2}\), utilize the power of a power rule: \((a^m)^n = a^{m\cdot n}\), yielding \(u^3\).
Combining these simplified results gives a final expression of \(8u^3\), showcasing how simplification reduces complexity. This practice is crucial for making calculations straightforward and for understanding larger mathematical problems.

Positive Exponents

Expressions are commonly required to be in the form of positive exponents, as it simplifies understanding and ensures consistency in expressing mathematical ideas.
An exponent represents the number of times a base is multiplied by itself. A positive exponent indicates regular multiplication, while negative exponents indicate division or reciprocal. Maintaining positive exponents ensures clarity, especially when communicating mathematical solutions.
When dealing with exponents, always aim for positive ones by simplifying negative or fractional exponents whenever possible. For example, in the expression \((4u^2)^{3/2}\), both numerical and variable exponents end up as positive results — \(4^{3/2}\) simplifies to 8, and \(u^3\) naturally is positive.
Mathematical results expressed in positive exponents are easier to interpret and are commonly required in academic evaluations and real-world applications.