Problem 2 - Intermediate Algebra

Question

We read \(16^{3 / 2}\) as " 16 to the three-_______ power.”

Step-by-Step Solution

Answer

"16 to the three-halves power."

Step 1: Understand fractional exponents

The expression \(16^{3/2}\) involves a fractional exponent. In general, when you have \(a^{m/n}\), it represents the \(n\)-th root of \(a\) raised to the power of \(m\). This means \(a^{m/n} = (\sqrt[n]{a})^m = \sqrt[n]{a^m}\).

Step 2: Identify the components

In the expression \(16^{3/2}\), the base is 16, the numerator of the exponent, 3, is the power to which the root result is raised, and the denominator of the exponent, 2, is the root that is taken. Thus, this expression can be interpreted as "The square root of 16 raised to the power of 3."

Step 3: Rewrite in words

Since the denominator of the exponent, \(2\), implies a square root, and the numerator, \(3\), implies cubing, we read \(16^{3/2}\) as "16 to the three-halves power."

Key Concepts

Square Root

The square root is a fundamental concept in mathematics. It helps us find a number which, when multiplied by itself, gives us the original number. For example, the square root of 16 is 4 because 4 times 4 equals 16. The square root symbol is denoted by \( \sqrt{} \). This means \( \sqrt{16} = 4 \).
When dealing with fractional exponents like \( 16^{3/2} \), the denominator (2 in this case) represents a square root. It tells us to take the square root of the base number, which is 16.

• \( \sqrt{16} = 4 \)
• In \( 16^{3/2} \), the 2 denotes the square root

Understanding square roots is crucial when simplifying expressions involving fractional exponents. It lays the foundation for more complex mathematical operations.

Exponentiation

Exponentiation is a mathematical operation that involves raising a number to a certain power. It is a way of expressing repeated multiplication of a number by itself. For example, \( 2^3 = 2 \times 2 \times 2 = 8 \). The number being multiplied is called the "base," and the power it is raised to is the "exponent."
In the expression \( 16^{3/2} \), exponentiation is used twice:

• First, the square root of 16 is found, which is exponentiation with a fraction: \( 16^{1/2} \).
• Second, the result is raised to the power of 3, which is a whole number exponent: \( (\sqrt{16})^3 \ = 4^3 \ = 64 \).

Exponents make it easier to write and work with very large or small numbers. They are vital in calculations involving fractional powers

Mathematical Notation

Mathematical notation is a system of symbols used to represent numbers and operations in mathematics. It allows us to express complex mathematical ideas in a concise and unambiguous way. For instance, the expression \( 16^{3/2} \) is a compact form that conveys a lot of information.
Each part of a notation has a specific meaning:

• The base (16) is the number being operated on.
• The numerator of the exponent (3) represents the power to which the base's root is raised.
• The denominator of the exponent (2) indicates the type of root to be taken.

Being comfortable with mathematical notation allows students to better understand and solve problems efficiently. It is like learning a language that enables clear mathematical communication.