Problem 89 - Beginning and Intermediate Algebra

Question

Simplify. $$ \frac{-6^{2}+4}{-2} $$

Step-by-Step Solution

Answer

The simplified result is 16.

Step 1: Evaluate the Exponent

Start by focusing on the exponent inside the expression. Calculate \(-6^2\). Here, the exponent only applies to 6 and not the negative sign; hence, calculate it as \((-6)^2 = 36.\)

Step 2: Replace Exponent with Result

Substitute \(36\) in place of \(6^2\) in the original expression. The expression becomes \(\frac{-36 + 4}{-2}.\)

Step 3: Simplify the Numerator

Simplify the numerator by performing the operation: \(-36 + 4 = -32\). Now the expression is \(\frac{-32}{-2}.\)

Step 4: Perform Division

Divide the simplified numerator by the denominator: \(-32 \div -2 = 16.\) Since negative signs cancel each other out, the result is positive.

Key Concepts

Exponents in Algebra

Exponents in algebra represent the number of times a number, called the base, is multiplied by itself. In the expression \(-6^2\), the base is 6, and the exponent is 2. This means you multiply 6 by itself, resulting in \(6 \times 6 = 36\). It is crucial to note that the exponent applies only to the number directly beside it unless brackets indicate otherwise.

For example, in the expression \((-6)^2\), the entire term \(-6\) is squared, yielding 36. However, in \(-6^2\), only 6 is squared, resulting in 36, but the negative sign remains outside. Therefore, \(-6^2 = -(6 \times 6) = -36\). You simply remember to treat the negative sign separately unless it's enclosed in parentheses. Understanding this is critical for accurately decoding and solving algebraic expressions involving exponents.

Division of Integers

Division of integers involves dividing one integer by another. In our exercise, the expression \( \frac{-32}{-2} \) is solved by dividing the negative integers.

When dividing two numbers, if both have the same sign, the result is positive. This is because negatives 'cancel out' when divided by each other. So, \(-32 \) divided by \(-2\) results in \(+16\).

Also, integers must be whole numbers, they can be negative or positive, but they cannot have decimal places. Practicing these simple rules helps maintain accuracy when working with division problems in algebra.

Arithmetic Operations

Arithmetic operations are basic mathematical procedures including addition, subtraction, multiplication, and division. Each operation follows specific steps to simplify an expression.

In the given problem, simplifying the numerator involves adding \(-36 + 4\). Here, you are performing an arithmetic operation of addition with negative numbers. By adjusting this with basic arithmetic rules, the result simplifies to \(-32\).

Combining operands requires adhering to the order of operations, commonly remembered as PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)). Following this ensures each operation is performed in the correct sequence. Practice calculating simple expressions using arithmetic operations to build a strong foundational understanding.