Problem 147 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(147-149,\) perform the indicated operation. $$(-6)^{2}=(-6)(-6)=?$$

Step-by-Step Solution

Answer

Therefore, the result of \((-6)^{2} = 36\).

Step 1: Understanding the Exponent

The equation is \((-6)^{2}.\) Here, the number -6 is raised to the exponent of 2. When a number is raised to the exponent of 2, it is being multiplied by itself.

Step 2: Performing the Multiplication

\((-6)(-6)\) will be calculated. According to the rule for multiplying two numbers of the same sign, the result is a positive number.

Step 3: Calculation

-6 multiplied by -6 equals 36.

Key Concepts

Negative Numbers Multiplication

Understanding how to multiply negative numbers is essential in algebra and beyond. The rule is relatively straightforward: when you multiply two negative numbers together, the result is positive. This might seem counterintuitive at first, but it's based on the idea that a negative number denotes the opposite of something. So, if we consider '-6' as 'the opposite of 6,' multiplying it by another '-6' (another 'opposite') cancels out the 'oppositeness', leaving us with a positive result.

For example, multiplying (-6) by (-6) results in 36, because each '-6' effectively represents the flip side of 6, and flipping something twice brings it back to its original state. This is just like saying that two wrongs make a right in the realm of multiplication. You can remember this rule with a simple phrase: 'a negative times a negative equals a positive.'

Exponents and Powers

Exponents and powers are a way of expressing repeated multiplication. The exponent, also known as the power, tells you how many times to multiply the base number by itself. It's a form of shorthand notation to make multiplication of the same number multiple times more efficient. In the expression (-6)^{2}, '-6' is the base and '2' is the exponent.

The exponent of 2 indicates that you should multiply -6 by itself once. So, the equation simplifies to (-6) * (-6). Here's where our rule about multiplying negative numbers comes into play again. Since both numbers are negative, the result will be a positive value. This leads us to a vital reminder: the exponent applies to the sign as well as the number, so be attentive to whether the negative is inside or outside the parenthesis—it makes a big difference.

Performing Multiplication

Performing multiplication, especially with negative numbers or more complex expressions, requires careful attention to rules and order. When faced with a multiplication problem, always start by identifying the numbers or expressions involved and take note of their signs. In our example, we have (-6) * (-6).

Here are steps to follow when performing such multiplication:

• Establish the rule for the signs involved—same signs yield a positive, and different signs yield a negative.
• Perform the multiplication of the numerical parts as you would with positive numbers.
• Combine the sign and numerical results to get the final answer.

In the case of (-6) * (-6), both numbers have the same sign. Thus, we multiply 6 by 6 to get 36 and attach a positive sign to the result, getting a final answer of 36.