Problem 49 - Intermediate Algebra

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 expression simplifies to \(8u^3\).

Step 1: Apply the Power of a Power Property

The expression \((4u^2)^{3/2}\) needs to be simplified using exponent rules. According to the power of a power property, \((a^m)^n = a^{m \cdot n}\). First, distribute the exponent \(\frac{3}{2}\) to both factors inside the parentheses: \(4^{3/2}\) and \( (u^2)^{3/2}\).

Step 2: Simplify the Numerical Base

Calculate the value of \(4^{3/2}\). This can be broken down into \((4^{1/2})^3\), which is equal to \(\sqrt{4}^3\). Since \(\sqrt{4} = 2\), the expression simplifies to \(2^3 = 8\).

Step 3: Simplify the Variable Base

Now simplify \((u^2)^{3/2}\) using the power of a power property: \((u^{2})^{3/2} = u^{2 \cdot (3/2)} = u^3\).

Step 4: Combine the Simplified Parts

Combine the simplified parts together: \(8u^3\). This is the expression with positive exponents.

Key Concepts

Power of a Power Property

When dealing with powers of powers, we use a specific rule to simplify expressions like \((a^m)^n\). The power of a power property states that you multiply the exponents together, so \((a^m)^n = a^{m \cdot n}\). This allows us to simplify complex exponential expressions by turning two sets of exponents into one.
For example, if you have \((u^2)^{3/2}\), the power of a power property tells us to multiply 2 by \(3/2\). This gives you the new exponent \(3\), simplifying the expression to \(u^3\).

• Always distribute the outer exponent to each factor inside the parenthesis.
• Be mindful of fractional exponents, as they often signify roots and can be simple to handle once you break them down.

Simplifying Expressions

Simplifying expressions involves applying well-known rules to make an expression shorter and easier to understand. When working with exponents, we aim to write our results using fewer powers and sometimes breaking down complex parts.
For example, in the expression \((4u^2)^{3/2}\), we first distribute the \(3/2\) to both \(4\) and \(u^2\). This means we end up with two separate expressions we need to simplify: \(4^{3/2}\) and \((u^2)^{3/2}\).

• Simplifying \(4^{3/2}\): Understand that a fractional exponent like \(3/2\) is the same as taking a square root, then cubing the result. So, \((\sqrt{4})^3 = 2^3 = 8\).
• Combining results: Next, combine \(8\) (from simplifying \(4^{3/2}\)) and \(u^3\) (from simplifying \((u^2)^{3/2}\)) to get \(8u^3\).

Positive Exponents

When tasked with simplifying expressions, we strive to end with positive exponents. Unlike negative or zero exponents, positive exponents describe the actual size or repetitions of multiplication for a number or variable.
In expressions like \(8u^3\), each exponent is positive, making it straightforward and easy to interpret. It tells us how many times each base is multiplied by itself—\(u^3\) means \(u\) is multiplied three times, giving a clear indication of its magnitude.

• Convert any negative exponents by finding their reciprocal if needed. But here, all exponents are already positive.
• Emphasizing positive exponents makes expressions elegant and their meaning clear, avoiding the need for further simplification.