Problem 20 - Brief Applied Calculus

Question

Evaluate each expression without using a calculator. $$ 16^{3 / 2} $$

Step-by-Step Solution

Answer

The value of \(16^{3/2}\) is 64.

Step 1: Understand the Expression

The given expression is \(16^{3/2}\). Here, the exponent \(\frac{3}{2}\) indicates a power and a root operation. The numerator \(3\) refers to raising the base to the power of 3, and the denominator \(2\) refers to taking the square root.

Step 2: Simplify the Expression with Root and Power

The expression can be rewritten as \((16^{1/2})^3\) because \(a^{m/n} = (a^{1/n})^m\) or as \((16^3)^{1/2}\) because \(a^{m/n} = (a^m)^{1/n}\). For simplicity, let's evaluate \((16^{1/2})^3\).

Step 3: Evaluate the Square Root

First, find \(16^{1/2}\), which is the square root of 16. We know that \(16 = 4^2\), so \(\sqrt{16} = 4\).

Step 4: Raise to the Power of 3

Now, take the result from Step 3 and raise it to the power of 3. Since \(16^{1/2} = 4\), we calculate \(4^3 = 4 \times 4 \times 4 = 64\).

Step 5: Verify with Power-First Approach

Alternatively, calculate \(16^3\) first and then take the square root. \(16^3\) is \(16 \times 16 \times 16 = 4096\). Taking the square root of 4096, we also get 64. This verifies our previous calculation.

Key Concepts

Square Root

A square root is a number which, when multiplied by itself, gives the original number. For example, the square root of 16 is 4 because when you multiply 4 by itself (4 x 4), you get 16. Learning to calculate square roots is fundamental in mathematics. Often, square roots are expressed using the radical symbol \( \sqrt{} \). In our scenario with \( 16^{3/2} \), recognizing \( 16^{1/2} \) as the square root of 16 helps break down the expression into simpler components.To calculate a square root:

• Identify the number whose root needs to be determined. For 16, this is straightforward because 16 is a perfect square.
• Look for a number which, when squared (multiplied by itself), equals the given number.

In mathematical calculations, understanding square roots paves the way for advanced concepts like rational exponents.

Rational Exponents

Rational exponents are an extension of traditional exponents and represent both power and roots within a single expression. When you encounter an exponent represented as a fraction, such as \( \frac{3}{2} \), it consists of a numerator and a denominator.

• The numerator indicates the power to which the base number should be raised. In this exercise, the 3 indicates that after taking the square root, the result should be cubed.
• The denominator represents the root that should be applied to the base. Thus, the \( \frac{1}{2} \) signifies the square root.

By utilizing rational exponents, mathematical operations can be unified in a single step, simplifying expressions like \( 16^{3/2} \) into either \( (16^{1/2})^3 \) or \( (16^3)^{1/2} \). This flexibility in representation allows us to strategically choose the easiest path to simplify and evaluate expressions.

Order of Operations

The order of operations is essential in mathematics for ensuring calculations are executed correctly. Following the proper order avoids confusion when dealing with multiple mathematical operations.When handling complex expressions, like \( 16^{3/2} \), we first address the root operation before moving on to the power operation. This is because the expression can be interpreted as \( (16^{1/2})^3 \):

• Start with the root: Calculate \( 16^{1/2} \), which equals 4, as it is the square root of 16.
• Proceed with the power: Then raise that result to the power of 3. So, \( 4^3 = 64 \).

This problem-solving method aligns with the standard mathematical principle known as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Emphasizing these steps ensures consistency and accuracy in mathematical evaluations.