Problem 2 - Elementary and 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, perfect fourth powers.

Step 1: Identify the Pattern 1

The numbers 1, 4, 9, 16, 25, and 36 can each be expressed as the square of an integer: \(1^2 = 1\), \(2^2 = 4\), \(3^2 = 9\), \(4^2 = 16\), \(5^2 = 25\), \(6^2 = 36\). Therefore, these numbers are called perfect squares.

Step 2: Identify the Pattern 2

The numbers 1, 8, 27, 64, and 125 each can be expressed as the cube of an integer: \(1^3 = 1\), \(2^3 = 8\), \(3^3 = 27\), \(4^3 = 64\), \(5^3 = 125\). Thus, these are called perfect cubes.

Step 3: Identify the Pattern 3

Numbers 1, 16, 81, 256, and 625 each can be expressed as the fourth power of an integer: \(1^4 = 1\), \(2^4 = 16\), \(3^4 = 81\), \(4^4 = 256\), \(5^4 = 625\). Hence, they are perfect fourth powers.

Key Concepts

Perfect cubes

A perfect cube is a number that can be expressed as an integer raised to the power of three. In simpler terms, it's the result you get when you multiply an integer by itself twice more. An example is the number 8, which is the cube of 2. The calculation goes as follows:

• Begin with the integer 2,
• Multiply it by itself: 2 x 2 = 4,
• Then, multiply the result (4) by the integer again: 4 x 2 = 8.

The integer that is raised to the power is called the "base." In the example above, the base is 2. Perfect cubes can also help you easily recognize patterns, especially when dealing with equations involving volume since cubes relate to three-dimensional space.

Exponents

Exponents represent a very efficient way of expressing repeated multiplication of the same number. They are written as a small number to the upper right of a base number. For example, the expression \(3^4\) means that the number 3 is multiplied by itself a total of four times: 3 x 3 x 3 x 3 = 81.
Exponents have laws that make calculations simpler:

• Multiplying similar bases: \(a^m \times a^n = a^{m+n}\),
• Dividing similar bases: \(a^m / a^n = a^{m-n}\),
• Power of a power: \((a^m)^n = a^{m\times n}\).

Understanding these laws is especially useful when solving algebraic equations or simplifying large numbers.

Number patterns

Number patterns are sequences that follow a predictable rule or formula. Recognizing these patterns can simplify complex math problems and help with understanding larger concepts. Take for example the sequence of numbers where each is a perfect square: 1, 4, 9, 16, 25, 36. Each number here follows the formula \(n^2\), showing that they are formed by squaring consecutive integers starting from 1.
By identifying the pattern, you gain an intuitive grasp of the sequence's behavior, which can be applied to various math problems:

• Arithmetic patterns, where each term after the first is obtained by adding a fixed number, are another common sequence type.
• Geometric patterns involve multiplying by a constant to get from one number to the next.

Understanding and spotting these patterns make solving math problems quicker and easier.