Problem 51 - College Algebra and Calculus: An Applied Approach

Question

Simplify the expression.\(\frac{2^{3 / 2}}{2}\)

Step-by-Step Solution

Answer

The simplified form of \(\frac{2^{3 / 2}}{2}\) is 1.414213562.

Step 1: Convert the exponent to the decimal form

First, convert the fractional exponent to a decimal to make the process easier. Hence, \(2^{3/2}\) becomes \(2^{1.5}\).

Step 2: Evaluate the exponent

The next step is to find the value of \(2^{1.5}\). By using a calculator, you find that \(2^{1.5}\) equals approximately 2.828427124.

Step 3: Perform the division

Finally, divide this result by 2, which yields 1.414213562.

Step 4: Re-check the original problem

At this point, check the result to see it matches the original problem. Simplifying \(\frac{2^{3/2}}{2}\) gives you the same result as the problem \(\frac{2^{1.5}}{2}\), so the work is correct.

Key Concepts

Fractional Exponents

Understanding fractional exponents is crucial in simplifying expressions. A fractional exponent like \(2^{3/2}\) represents both a power and a root. The top number (numerator) of the fraction is the power, while the bottom number (denominator) is the root. In this case, \(2^{3/2}\) means the cube of the square root of 2.
This can be transformed into two steps:

• First, find the square root of 2
• Then, raise the result to the power of 3

To skip intermediate steps, \(2^{3/2}\) can be simplified directly using a calculator by calculating \(2^{1.5}\), leading to approximately 2.828.
Understanding this concept helps in deciphering more complex expressions involving fractional powers.

Division

Division is a fundamental operation that involves determining how many times one number fits into another. In the context of exponents, division serves as a balancing scale, especially when dealing with terms that share the same base.
For example, in the expression \(\frac{2^{3/2}}{2}\), you are essentially dividing \(2^{1.5}\) by \(2^1\). With division, you subtract the exponents: \(3/2 - 1 = 1/2\). This results in \(2^{1/2}\), which simplifies to the square root of 2.
Thus, the final result is approximately 1.414.
Recognizing how division interacts with exponents can simplify complex expressions significantly.

Exponentiation

Exponentiation refers to the process of raising a number to a power. This operation magnifies or diminishes a number depending on the power applied.
When working with exponents, it is important to note the rules:

• Multiplying Like Bases: Add the exponents \((a^m \times a^n = a^{m+n})\)
• Dividing Like Bases: Subtract the exponents \((a^m / a^n = a^{m-n})\)
• Powers of a Power: Multiply the exponents \(( (a^m)^n = a^{m \cdot n})\)

Understanding these fundamental rules helps in managing expressions effectively. In \(2^{3/2}\), we raised 2 to the 1.5th power to deal with fractional exponents, essential for solving the initial division expression.

Algebraic Operations

Algebraic operations encompass the basic arithmetic functions: addition, subtraction, multiplication, and division. These operations extend to more advanced procedures when incorporating variables, fractions, and exponents.
When simplifying an expression, the goal is to reduce it to its simplest form. This often involves using multiple algebraic operations in tandem. For instance, the exercise \(\frac{2^{3/2}}{2}\) involves exponentiation first, followed by division.
Each operation builds upon the previous one with a consistent process vocabulary aiding the simplification. Grasping the methods of handling numbers in expressions will allow building proficiency in algebra, better equipping to tackle complex problems.