Problem 58 - Algebra 1

Question

Simplify the variable expression. $$(-5)^{2}(-y)(-y)$$

Step-by-Step Solution

Answer

The original variable expression simplifies to \(25y^2\).

Step 1: Evaluate exponential operation on constant.

The first operation to be performed is exponential operation on the constant \(-5^2\), which evaluates to \(25\). Therefore, \(-5^{2}(-y)(-y)\) becomes \(25(-y)(-y)\).

Step 2: Apply the rule for multiplication of negative numbers.

Next, apply the rule for multiplication of negative numbers which states that the product of two negative numbers is positive. Therefore, \((-y)(-y)\) become \(y^2\). Now, our expression \(25(-y)(-y)\) turns to \(25y^2\)

Step 3: Simplify the product

Finally, multiply the constant \(25\) with the variable \(y^2\) to obtain \(25y^2\), which is the simplified form of the original expression.

Key Concepts

Exponentiation

Exponentiation is a mathematical operation that involves raising a number, known as the base, to the power of an exponent. For instance, in the expression \(a^n\), \(a\) represents the base and \(n\) the exponent, indicating that \(a\) is to be multiplied by itself \(n\) times. When we encounter \(a^2\), it's the same as saying \(a \cdot a\). In the given exercise, \( (-5)^2 \) represents exponentiation, where -5 is the base and 2 is the exponent, resulting in \( (-5) \cdot (-5) = 25 \). Understanding exponentiation is crucial when simplifying variable expressions, as it allows us to compute the powers of numbers before dealing with other operations.

Multiplication of Negative Numbers

Grasping the concept of multiplying negative numbers is essential in algebra. The rule states that when two negative numbers are multiplied together, their product is positive. This is because a negative times a negative results in a positive. To visualize why this is the case, consider that a negative number can be thought of as 'opposite direction'. Thus, if we say 'opposite of opposite', it's akin to 'right back where we started' - which is positive. This concept applies directly to our exercise, since we multiply two negative numbers \( (-y)(-y) \) to get a positive result \( y^2 \). Keeping this rule in mind is vital for correctly simplifying expressions involving negative numbers.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operations (such as addition, subtraction, multiplication, and division). When simplifying algebraic expressions, we follow the order of operations, known as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). However, it's not just about the mechanical process of simplification; understanding how variables interact through these operations is key. In the solution we examined, the expression \(25(-y)(-y)\) ultimately simplifies to \(25y^2\). Here, the variable \(y\) retains its identity through the multiplications, and understanding how \(y\) behaves in different mathematical scenarios allows for more fluid manipulation of algebraic expressions. Being comfortable with algebraic expressions enables students to solve more complex equations with confidence.