Problem 6 - A Gentle Introduction to the Art of Mathematics

Question

Prove that the set of perfect fourth powers is contained in the set of perfect squares.

Step-by-Step Solution

Answer

A perfect fourth power \( a^4 \) can be expressed as \( (a^2)^2 \), a perfect square.

Step 1: Understand the Problem

The objective is to prove that any perfect fourth power is also a perfect square. A perfect fourth power is a number of the form \( n^4 \) while a perfect square is of the form \( m^2 \).

Step 2: Express a Perfect Fourth Power

Write a number that is a perfect fourth power in its mathematical form: \( a^4 \), where \( a \) is some integer.

Step 3: Recognize a Property of Exponents

Recall that the power of a power rule states \( (a^m)^n = a^{m \times n} \). So, \( a^4 = (a^2)^2 \).

Step 4: Identify the Perfect Square

Observe that \( a^4 = (a^2)^2 \) is in the form \( b^2 \) with \( b = a^2 \). Therefore, \( a^4 \) is a perfect square.

Step 5: Conclude the Proof

Since any perfect fourth power \( a^4 \) can be written as a square of \( a^2 \), which is itself an integer, it is true that every perfect fourth power is contained in the set of perfect squares.

Key Concepts

Perfect Fourth Power

Let's start by understanding what a perfect fourth power is. A number is called a perfect fourth power if it can be expressed as the fourth power of an integer. In mathematical terms, if 'a' is an integer, then a number of the form \(a^4\) is a perfect fourth power.

For example:

• \(2^4 = 16\)
• \(3^4= 81\)
• \(4^4 = 256\)

These are all perfect fourth powers because 16, 81, and 256 can be written as the fourth powers of integers 2, 3, and 4 respectively.

Perfect Squares

Next, we need to understand what a perfect square is. A perfect square is a number that can be expressed as the square of an integer. In other words, if 'b' is an integer, then a number of the form \(b^2\) is a perfect square.

Examples of perfect squares include:

• \(2^2 = 4\)
• \(3^2 = 9\)
• \(4^2 = 16\)

These are perfect squares because 4, 9, and 16 can be written as the squares of integers 2, 3, and 4 respectively.

Properties of Exponents

One essential property of exponents that helps with these kinds of proofs is the power of a power rule. The power of a power rule states that \((a^m)^n = a^{m \times n}\). This means that when you raise a power to another power, you multiply the exponents.

Let's apply this property to our example: we have \(a^4\) and we want to prove it’s a perfect square. Using the power of a power rule:

• \(a^4 = (a^2)^2\)

By rewriting \(a^4\) as \((a^2)^2\), we see that \(a^2\) is an integer and squaring it results in a perfect square. Hence, \(a^4\) can be written as the square of \(a^2\), proving \(a^4\) is indeed a perfect square.

Integer

To wrap things up, it's important to pinpoint the role of integers in this proof. An integer is a whole number that can be positive, negative, or zero without any fractions or decimals.

When we talk about perfect fourth powers and perfect squares, we are specifically referring to integers. For example, in our proof, we expressed \(a^4\) with 'a' being an integer. When we rewrote this as \((a^2)^2\), \(a^2\) remains an integer because squaring an integer always results in another integer.

Hence, dealing with integers helps ensure that these transformations and properties hold true, solidifying the proof that every perfect fourth power is a perfect square.