Problem 55 - Fundamentals of Mathematics

Question

Determine the value of each of the powers. Use a calculator to check each result. \(42^{2}\)

Step-by-Step Solution

Answer

The value of \(42^2\) is 1764.

Step 1: Identify the Base and Exponent

The expression given is \(42^2\). In this expression, \(42\) is the base and \(2\) is the exponent.

Step 2: Expand the Expression

Since the exponent is \(2\), which means square, we expand the expression as \(42 \times 42\).

Step 3: Perform the Multiplication

Now, multiply \(42\) by \(42\).
\(42 \times 42 = 1764\).

Step 4: Verify With a Calculator

Use a calculator to type in \(42^2\) or \(42 \times 42\) to confirm that the result is indeed \(1764\).

Key Concepts

Base and Exponent

In mathematics, expressions involving exponents are an essential area to understand. When you see an expression like \(42^2\), it consists of a base and an exponent. Here, the number \(42\) is known as the base. The base is the number that will be multiplied by itself.
The exponent, in this case **2**, indicates how many times the base is used in the multiplication. For instance, if the exponent is 2, it means we multiply the base once by itself.
In general, if you have a base \(a\) and an exponent \(n\), the expression can be written as \(a^n\), meaning \(a\) multiplied by itself \(n\) times. Understanding this concept is crucial as it simplifies expression evaluation and application in various mathematical problems.

Key points to remember:

• The base is the number being multiplied.
• The exponent tells you how many times to multiply the base by itself.
• This format is commonly used in algebra, geometry, and calculus.

Squaring Numbers

Squaring numbers is a specific type of operation where the exponent is 2. It is essential in different branches of mathematics and appears frequently in algebra and geometry.
When a number is squared, it is multiplied by itself. In our example, \(42^2\), we multiply 42 by 42. This action results in \(42 \times 42 = 1764\).
This operation can visualize geometric concepts as well. For example, squaring can represent the calculation of an area of a square, where one side length is the base. If a square has sides of length 42, its area is \(1764\) square units.

Some points to consider include

• Squares are always positive numbers since a negative number times itself results in a positive.
• Squares of whole numbers are called perfect squares (e.g., \(4, 9, 16\)).
• Squaring is frequently used in formulas for physics and engineering calculations.

Mathematical Multiplication

Mathematical multiplication is a fundamental operation where numbers are combined to form a new, larger number called the product. Understanding multiplication is essential not only in squaring numbers but also in various everyday calculations and more advanced math concepts.
When multiplying, you add a number to itself a certain number of times. For instance, \(42 \times 42\) can be thought of as adding 42 to zero, 41 more times.
The process of multiplication can be executed manually or with tools like calculators for efficiency, especially when dealing with large numbers.
Some techniques to understand multiplication better include

• Breaking numbers into components (e.g., \(42 \times 42 = (40+2)\times(40+2)\)).
• Employing the distributive property: \((a+b)(c+d)=ac+ad+bc+bd\).
• Memorizing multiplication tables to improve speed and confidence in solving problems.

Mastering multiplication lays the foundation for diving deeper into complex mathematical concepts.