Problem 27 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Carry out the indicated operation and write your answer using positive exponents only. $$ \left(2^{3 / 2}\right)^{4} $$

Step-by-Step Solution

Answer

The short answer to the question is: \(\left(2^{3 / 2}\right)^{4} = 2^{6}\).

Step 1: Identify the base and exponents

Given expression: \(\left(2^{3 / 2}\right)^{4}\) Here, the base is \(2\) and the exponents are \(\frac{3}{2}\) and \(4\).

Step 2: Apply the exponent rule

Apply the exponent rule \((a^m)^n = a^{m*n}\) to the expression: \(\left(2^{3/2}\right)^4 = 2^{(3/2)*4}\)

Step 3: Simplify the exponent

Simplify the exponent by multiplying the fractions: \(2^{(3/2)*4} = 2^{(3*4)/2}\)

Step 4: Evaluate the final expression

Evaluate the expression with the simplified exponent: \(2^{(3*4)/2} = 2^6\) Now our answer is in a positive exponent form, which is the desired format: \(\left(2^{3 / 2}\right)^{4} = 2^{6}\)

Key Concepts

Exponent Rules

Exponent rules are essential for simplifying expressions that involve powers. These rules help in manipulating and solving expressions where numbers are raised to powers. Understanding these rules simplifies how we deal with exponents.

One key rule is the power of a power property, which states

• \((a^m)^n = a^{m \times n}\)

This means we multiply the exponents together when raising a power to another power. For example, in the expression \((2^{3/2})^4\), applying this rule results in combining

• the exponents \(3/2\) and \(4\),

which results in the new exponent \((3/2) \times 4\). By mastering this exponent rule, you can easily simplify such expressions and handle more complex problems. It's crucial for solving problems accurately and efficiently.

Positive Exponents

Positive exponents signify that you multiply the base a certain number of times. They are straightforward to deal with, unlike negative exponents that indicate division or fractional components.

The ultimate goal is often to express results using positive exponents, which are more intuitive. In the context of the given problem, after performing the operation, the expression \((2^{3/2})^4\) simplifies to \(2^6\). Here, 6 is a positive exponent, which indicates that \(2\) is multiplied by itself six times:

• \(2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64\).

Using positive exponents makes the expression clearer and easier to interpret. Students are often required to express answers in positive exponent form for clarity and to standardize mathematical solutions.

Simplifying Exponents

Simplifying exponents involves reducing expressions using exponent rules to make calculations easier.

In the exercise, we apply the multiplication of exponents property: \((a^m)^n = a^{m*n}\). For example, simplifying \((2^{3/2})^4\) results in \(2^{6}\). Here's how it happens step-by-step:

• First, multiply the exponents \(3/2\) and \(4\).
• Then, simplify the fraction: \((3 \times 4)/2 = 6\).
• The expression \((2^{3/2})^4\) simplifies to \(2^6\).

This simplification is key in algebra, where combining like terms and making expressions shorter are often necessary. Simplified forms are not only easier to manage but often required in exams to ensure complete understanding of exponential expression manipulation.