Problem 6 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Rewrite the number without using exponents. $$ -4^{2} $$

Step-by-Step Solution

Answer

The expression \(-4^2\) can be rewritten without using exponents as follows: \((-4)^2 = -4 \times -4 = 16\). So, the simplified expression is 16.

Step 1: Identify the base and the exponent

The base is -4, and the exponent is 2. The expression can be simplified as: \((-4)^2 = -4 \times -4\)

Step 2: Perform the multiplication

Multiply -4 by itself (-4 multiplied by -4): \((-4) \times (-4) = 16\)

Step 3: Write the simplified expression

The simplified expression without using exponents is 16: \(-4^2 = 16\)

Key Concepts

Exponents

Exponents are a shorthand way of expressing repeated multiplication. When you see something written like \( -4^2 \), this means you are multiplying the base, \(-4\), by itself.

Here's how exponents work:

• The number being multiplied is known as the "base." In our example, \(-4\) is the base.
• The exponent, which appears as a superscript, tells us how many times to multiply the base by itself. In \(-4^2\), the exponent is 2, which means you multiply \(-4\) times \(-4\).

Understanding exponents as repeated multiplication is essential. It simplifies calculations and makes it easier to work with large numbers or repeated operations. Instead of writing a multiplication expression several times, you can simply use an exponent for efficiency and clarity.

Multiplication

Multiplying numbers, especially when dealing with negative numbers, is a foundational arithmetic skill. When you multiply two numbers, you are essentially adding one number to itself as many times as specified by the other number.

In the expression \( (-4) \times (-4) \), we're multiplying the number \(-4\) by itself:

• Start by considering the absolute values of the numbers involved; in this case, 4 and 4.
• 4 times 4 equals 16.

However, since both numbers in our example are negative, multiplying them together results in a positive product. Remember this specific rule about multiplication:

• A negative times a negative always gives a positive result.
• A positive times a positive gives a positive result.
• A positive times a negative (or vice versa) gives a negative result.

Negative Numbers

Negative numbers extend the number line to values less than zero, allowing for a greater range of mathematical operations. These numbers have a few key properties, especially when involved in operations such as exponentiation and multiplication.

Here's what you need to know about negative numbers:

• Negative numbers are denoted with a minus sign (-) in front of them, as seen in \(-4\).
• When you square a negative number, like \((-4)^2\), multiply the number by itself; however, treat the operation as if the parentheses enforce the negative with each multiplication. Thus, \(-4 \times -4 = 16\).
• An even exponent of a negative number results in a positive because the negative signs "cancel out." But an odd exponent would result in a negative product—like \(-4^3 = -4 \times -4 \times -4 = -64\).

Negative numbers are versatile and can drastically change the results of calculations when combined with an exponent or when multiplied. Understanding their properties is key to mastering algebra and other branches of mathematics involving these concepts.