Problem 2 - Intermediate Algebra

Question

Fill in the blanks. Numbers such as \(1,4,9,16,25,\) and 36 are called perfect___. Numbers such as \(1,8,27,64,\) and 125 are called perfect___. Numbers such as \(1,16,81,256,\) and 625 are called perfect-fourth ___.

Step-by-Step Solution

Answer

Perfect squares, perfect cubes, and perfect fourth powers.

Step 1: Recognize Perfect Squares

Look at the first sequence of numbers: 1, 4, 9, 16, 25, and 36. Notice that these are the squares of whole numbers, namely 1 (since \(1^2 = 1\)), 2 (since \(2^2 = 4\)), 3 (since \(3^2 = 9\)), 4 (since \(4^2 = 16\)), 5 (since \(5^2 = 25\)), and 6 (since \(6^2 = 36\)). These numbers are called perfect squares.

Step 2: Identify Perfect Cubes

Consider the second sequence: 1, 8, 27, 64, and 125. These numbers can be expressed as cubes of integers, namely 1 (since \(1^3 = 1\)), 2 (since \(2^3 = 8\)), 3 (since \(3^3 = 27\)), 4 (since \(4^3 = 64\)), and 5 (since \(5^3 = 125\)). Therefore, these numbers are recognized as perfect cubes.

Step 3: Determine Perfect Fourth Powers

For the third sequence: 1, 16, 81, 256, and 625, these numbers are results of raising integers to the fourth power. For example, 1 (since \(1^4 = 1\)), 2 (since \(2^4 = 16\)), 3 (since \(3^4 = 81\)), 4 (since \(4^4 = 256\)), and 5 (since \(5^4 = 625\)). As such, they are called perfect fourth powers.

Key Concepts

Perfect Squares

Perfect squares are numbers that result from squaring a whole number. Imagine a square garden; each side length is the same. When you calculate the total area by multiplying one side by itself, you get a perfect square.
For instance, if one side is 3 units, then the area is \( 3^2 = 9 \).
This pattern applies to all perfect squares:

• 1 because \( 1^2 = 1 \)
• 4 because \( 2^2 = 4 \)
• 9 because \( 3^2 = 9 \)
• 16 because \( 4^2 = 16 \)

These numbers, like many others, form the sequence of perfect squares and are critical in various mathematical calculations and geometric interpretations.

Perfect Cubes

Perfect cubes are numbers that you obtain when you raise a whole number to the power of three. Picture a three-dimensional box (a cube) where the length, width, and height are all the same. When you multiply one side three times, it gives you the volume of the cube—a perfect cube.
To demonstrate:

• By raising 1, you get \( 1^3 = 1 \)
• For 2, you have \( 2^3 = 8 \)
• With 3, it's \( 3^3 = 27 \)
• And 4 results in \( 4^3 = 64 \)

These figures belong to the collection of perfect cubes, fundamental to understanding three-dimensional space.

Perfect Fourth Powers

Perfect fourth powers arise when you raise a whole number to the power of four. It’s like taking a number and multiplying it by itself four times. While not as frequently discussed as squares and cubes, fourth powers still play an essential role in mathematics.
Consider these computations:

• 1 raised to the fourth is \( 1^4 = 1 \)
• 2 to the fourth is \( 2^4 = 16 \)
• 3 to the fourth equals \( 3^4 = 81 \)
• 4 to the fourth results in \( 4^4 = 256 \)

These numbers are examples of perfect fourth powers. Recognizing these can be crucial for certain advanced calculations in algebra and number theory.