Problem 38 - Elementary Algebra Within Reach

Question

$$ \text { In Exercises 33-46, simplify the expression. } $$ $$ (-3 y)(-4 y) $$

Step-by-Step Solution

Answer

The simplified expression of (-3 y)(-4 y) is \(12y^2\) .

Step 1: Analyzing the Signs

First, we need to look at the signs of numbers and variables involved in the multiplication process. If the signs of both integers are negative then the sign of the result should be positive.

Step 2: Numerical multiplication

Consequently, multiplying the numerical parts i.e. -3 and -4, the result would be positive 12 due to the negative-negative rule in mathematics.

Step 3: Variable multiplication

When it comes to variables, we just have to multiply \(y\) and \(y\) to get \(y^2\). Basically, the rule of multiplying variables means adding their powers (exponents). In this case, both \(y\)'s have an implicit power of 1. Hence \(y \cdot y = y^{1+1} = y^2\).

Step 4: Combine the results

Finally, after multiplying both numerical and literal parts separately, combine these two answers together to get the final result. Thus, the final answer is \(12y^2\) .

Key Concepts

Negative-Negative Rule in Mathematics

Understanding the negative-negative rule in mathematics is crucial when simplifying algebraic expressions. This rule states that the product of two negative numbers or terms is a positive number. Imagine if you've had a double cancellation of a debt; two negatives have made a positive impact on your finances! In algebra, this concept is just as relieving and useful.

Let's consider the expression \( (-3) \times (-4) \). Both \( -3 \) and \( -4 \) are negative numbers. According to the negative-negative rule, if we multiply them together, we get a positive result because a negative times a negative gives a positive. Hence, \( (-3) \times (-4) = 12 \). This results in simplifying our initial problem by eliminating the complexity of dealing with negative signs early on in the process. What seemed daunting at first is now simpler and more manageable thanks to this fundamental rule in mathematics.

Exponents in Algebra

Exponents in algebra are shorthand for repeated multiplication of the same number or variable. When you see a term raised to a power, it signals how many times that term is to be used in a multiplication. For example, \( y^2 \) is the same as saying \( y \times y \).

When multiplying variables with the same base, we add their exponents. This is based on the laws of exponents, which streamline our algebraic work. If you encounter \( y \times y \), you are dealing with two variables of the same base (\( y \) in this case) both implicitly raised to the power of 1 (since \( y \) is the same as \( y^1 \)). The law of exponents tells us to add these powers together, resulting in \( y^1 + y^1 = y^2 \). This simplifies the multiplication of variables, allowing us to efficiently combine like terms and reduce the expression to its most concise form.

Multiplication of Algebraic Terms

Multiplying algebraic terms involves combining the numerical coefficients (numbers in front of the variables) and the variables themselves. The process is two-pronged: multiply the coefficients together and then multiply the variables together.

Consider the expression \( (-3y) \times (-4y) \). The coefficients here are \( -3 \) and \( -4 \) and the variable is \( y \). First, apply the negative-negative rule to multiply the coefficients: \( (-3) \times (-4) = 12 \). Next, focus on the variables: each \( y \) comes with an implicit exponent of 1. According to the laws of exponents, \( y^1 \times y^1 = y^{1+1} = y^2 \).

Finally, combine the results of the two separate multiplications (coefficients and variables): \( 12 \times y^2 = 12y^2 \). This final expression is simplified because it combines like terms effectively, resulting in a more straightforward and comprehensible expression.