Problem 37 - Algebra 1: Concepts and Skills

Question

Simplify the expression. \(-2(5)(-r)(-r)\)

Step-by-Step Solution

Answer

The simplified expression is -10r^2.

Step 1: Multiplication of Numerical Values

Start by multiplying the numerical values. Here, this means multiply -2 and 5.

Step 2: Follow Rule of Signs

The rule of signs states that multiplying two negative numbers gives a positive number. Therefore, \(-r*(-r)\) is \(r^2\).

Step 3: Put Together

Lastly, we put together the results. Multiply the result from step 1 with the result from step 2.

Key Concepts

Multiplication of Numerical Values

When multiplying numerical values, we directly multiply the numbers involved. Consider the expression \from our given exercise, -2 and 5 are numerical values. To multiply these, we simply calculate \to get -10. It's essential to remember that when we multiply a positive number by a negative number, the result is always negative. This is a fundamental principle that helps simplify expressions before we even deal with the variables involved.

Multiplication is commutative, meaning that the order in which we multiply numbers does not affect the result. For example, \[(-3) \times 4 = 4 \times (-3) = -12\]. Regardless of the order, the product remains the same. Ensuring we have the correct sign for our numerical product sets the stage for combining it with any variables in the expression.

Rule of Signs

The rule of signs is crucial for understanding how to simplify expressions involving multiplication and division with signed numbers. When multiplying two numbers with the same sign, the result is always positive: \[(-a) \times (-b) = ab\]. Conversely, if the signs are different, the result is negative: \[(-a) \times b = -(ab)\] and \[a \times (-b) = -(ab)\].

In the context of our exercise, we apply this rule to \[(-r) \times (-r)\], which are two negative factors. According to the rule of signs, the product will be positive, giving us \[r^2\]. This step simplifies the expression significantly and demonstrates the power of understanding and applying the rule of signs in algebra.

Exponentiation in Algebra

Exponentiation in algebra is a form of mathematical shorthand for expressing repeated multiplication of the same factor. When we have \[a \times a\], it's equivalent to \[a^2\], which reads as '\to the second power' or 'squared'. It's vital to distinguish between exponentiation and other operations, as it strictly implies multiple instances of the same base number being multiplied together.

For the term \[(-r) \times (-r)\] in our exercise, we've learned from the rule of signs that multiplying by a negative by itself results in a positive outcome. Thus \[(-r) \times (-r) = r^2\]. Here, we have exponentiated the base \(r\), indicating that \(r\) is used as a factor twice. Recognizing this expedites the simplification process, helping students understand the power of exponentiation as a tool for simplifying algebraic expressions.