Problem 89 - Algebra A Combined Function

Question

Evaluate each expression. \(\frac{-6^{2}+4}{-2}\)

Step-by-Step Solution

Answer

The value of the expression is -20.

Step 1: Apply Exponentiation

Begin with the exponent in the expression. Here, you have \(-6^2\). This means you multiply \(-6\) by itself. \(-6 \times -6 = 36\)

Step 2: Evaluate the Numerator

Substitute the result back into the expression:\(\frac{36 + 4}{-2}\).Next, solve for the numerator:\(36 + 4 = 40\)

Step 3: Simplify the Fraction

Now that the numerator is simplified to 40, you can substitute back to simplify:\(\frac{40}{-2}\) Divide 40 by -2:\(40 \div -2 = -20\)

Key Concepts

Exponentiation

Exponentiation is a mathematical operation that involves raising a number, called the base, to the power of another number, called the exponent. This means repeated multiplication of the base by itself. In our original exercise, we encounter the expression \(-6^2\). Here, the base is \-6\, and the exponent is 2.
It's important to note that when an exponent is applied, it means multiplying the base by itself. Hence, \(-6 \times -6 = 36\).

• If the base is negative and the exponent is even, the result is positive, which is why \(-6^2\) gives us 36.
• If the base is negative and the exponent is odd, the result remains negative.
• Always perform the exponentiation operation before any other operation in an expression.

Understanding and correctly applying exponentiation is crucial, as it can significantly alter the outcome of mathematical expressions.

Order of Operations

Order of operations is a foundational principle in mathematics that ensures expressions are solved consistently and correctly. Often remembered by the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).
In our expression \( \frac{-6^2+4}{-2}\), we must first address the exponentiation \(-6^2\), which we simplified to 36. Then, evaluate the addition in the numerator: \(36 + 4\).
After simplifying the numerator, division follows in the denominator.

• Always start with operations inside parentheses or other grouping symbols.
• Evaluate exponents before moving to multiplication, division, addition, or subtraction unless guided by parentheses.
• Multiplication and division are performed before addition and subtraction, and all operations of the same rank are handled from left to right.

Following this order accurately will ensure clarity and correctness in evaluating expressions.

Evaluating Expressions

Evaluating expressions involves computing the value of a mathematical expression by following established rules and principles, like the order of operations. In the given problem, after simplifying \(-6^2\) to get 36, we computed the entire numerator \(36 + 4\) to get 40.
These evaluations are essential for resolving algebraic expressions correctly. Once the numerator is calculated, we then move on to evaluate the fraction, dividing the numerator by the denominator.
To make this process smoother:

• Break down the expression into smaller parts and solve each part step-by-step, focusing on one operation at a time.
• Ensure each operation respects the rules for handling numbers, especially when negative numbers or exponents are involved.
• Double-check each step to minimize errors and maintain clarity.

Evaluating expressions requires attention to detail and clear understanding of mathematical processes.

Negative Numbers

Negative numbers are values less than zero and are often indicated by a minus (-) sign. They're crucial in algebra and require careful handling, especially when used in operations like multiplication or division.
In our example, we see negative numbers in both the base of the exponentiation \(-6\) and the denominator \(-2\). Multiplying or dividing by a negative number affects the sign of the result.
Here's a quick guide to handling negatives:

• Multiplying or dividing two negative numbers results in a positive number.
• Multiplying or dividing a positive number by a negative number results in a negative number.
• Watch out for negative signs in expressions; they can change the whole outcome if not dealt with correctly.

Properly dealing with negative numbers ensures accuracy and prevents common errors in algebraic calculations.