Problem 90 - Algebra 1

Question

Evaluate the expression. y^{2}-y \text { when } y=-2

Step-by-Step Solution

Answer

The value of the expression \(y^{2}-y\) when \(y=-2\) is 6.

Step 1: Substitute the value of y into the expression

Replace \(y\) in the expression \(y^{2}-y\) with -2. After this substitution, our expression would look like: \((-2)^{2} - (-2)\).

Step 2: Apply the order of operations

We must follow the order of operations, which says we should calculate the exponent before doing subtraction. When we square -2, we get 4, and -(-2) gives us 2. So, the expression becomes: \(4 + 2\).

Step 3: Perform the addition

Now we perform the addition operation to find the final answer: \(4 + 2 = 6\).

Key Concepts

Order of Operations

Understanding the order of operations is crucial when solving mathematical expressions. It ensures that everyone calculates expressions consistently and arrives at the same result. The universally accepted acronym for remembering the correct order is PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
In the given problem, after substituting the value of \(y\), the expression becomes \((-2)^2 - (-2)\). Following PEMDAS, we first address the exponentiation, \((-2)^2\), before moving on to other operations. This process not only aids in achieving the correct answer but also in establishing a strong foundation in mathematics.
By adhering to the order of operations, you maintain clarity and accuracy in your calculations, which is essential for any complex mathematical problem. Always remember to look for exponents and multiplication/division tasks before jumping into addition or subtraction.

Exponentiation

Exponentiation is a mathematical operation involving numbers called the base and the exponent or power. It indicates how many times the base is multiplied by itself. For example, in the expression \((-2)^2\), -2 is the base, and 2 is the exponent, signifying that -2 is multiplied by itself once: \((-2) \times (-2)\). This equals 4.
It's critical to handle negative bases carefully. When a negative number is raised to an even power, the result is positive because multiplying two negative numbers yields a positive product. Conversely, raising a negative number to an odd power results in a negative product. This understanding helps eliminate mistakes and confusion during calculations.

• For even exponents: negative base results in a positive number
• For odd exponents: negative base remains negative

Grasping this aspect of exponentiation ensures precision when solving problems that involve powers.

Arithmetic Operations

Arithmetic operations are the basic building blocks of math and include addition, subtraction, multiplication, and division. In this exercise, the operations focus primarily on subtraction and addition, both of which are performed after applying the order of operations and exponentiation.
Once the exponentiation of \((-2)^2\) occurred resulting in \(4\), and -(-2) was evaluated as \(+2\), the expression simplified to \(4 + 2\). These arithmetic operations become straightforward once previous steps are thoroughly completed.
Subtraction can often include handling of negative numbers, as seen when \(-(-2)\) transformed into \(+2\). Hence, understanding how negative signs affect arithmetic operations is vital. With addition, the operation simply sums values, providing the final result in the given problem.

• Adding positive and negative numbers requires careful attention to signs
• Combining arithmetic operations efficiently resolves the expression
  

Mastering these operations paves the way for more advanced mathematics, as they are essential in simplifying and solving expressions.