Problem 92 - Algebra A Combined Function

Question

Evaluate each expression. \(\frac{-2-4^{2}}{3(-6)}\)

Step-by-Step Solution

Answer

The expression evaluates to 1.

Step 1: Evaluate the Exponent

First, calculate the value of the exponent in the numerator. The expression inside the fraction is \(-2 - 4^2\). Calculate \(4^2\) which is \(16\).

Step 2: Simplify the Numerator

Replace \(4^2\) in the expression with its value: \(-2 - 16\). Now, perform the subtraction: \(-2 - 16 = -18\). The numerator is now \(-18\).

Step 3: Calculate the Denominator

In the denominator, you have \(3(-6)\). Perform the multiplication: \(3 imes -6 = -18\). The denominator is \(-18\).

Step 4: Perform the Division

Now, divide the simplified numerator by the simplified denominator: \(-18) \div (-18) = 1\).

Step 5: Final Answer

The expression evaluates to \(1\).

Key Concepts

Exponents in Algebra

Exponents in algebra signify repeated multiplication of a number by itself. Let’s consider the notation \( 4^2 \), also read as “4 squared.” This means the number 4 is multiplied by itself once, resulting in \( 4 \times 4 \). Therefore, \( 4^2 = 16 \). When dealing with exponents in algebraic expressions, it is essential to evaluate these components before performing other operations. This ensures accuracy in your calculations and is particularly crucial when you have negative bases or exponents, as these can alter the sign or the magnitude of the result. Always remember, when calculating exponents, use the base number as directed by the exponent and multiply it by itself the number of times the exponent specifies.

Numerator and Denominator

In any fraction, the number above the dividing line is called the numerator, while the number below it is the denominator. These components form the foundational structure for fractions, and understanding them is key to many algebraic processes. For example, in the fraction \( \frac{-18}{-18} \), \(-18\) is the numerator, and \(-18\) is the denominator. The numerator represents how many parts we are considering, and the denominator indicates how many parts make up a whole.

• The numerator can be a result of various arithmetic operations, including addition, subtraction, multiplication, and division.
• The same applies to the denominator.

Once both components are simplified, the fraction represents the relationship between these two numbers. A crucial point to note is the need for accurate simplification of both the numerator and denominator, which leads to the correct simplification of the fraction.

Order of Operations

The order of operations is a set of rules that dictate the sequence in which certain operations are performed to ensure consistent and correct results. A widely known acronym to remember this order is PEMDAS:

• P: Parentheses – Solve expressions inside parentheses first.
• E: Exponents – Next, evaluate exponents including powers and roots.
• MD: Multiplication and Division – Perform these operations from left to right as they appear in the expression.
• AS: Addition and Subtraction – Finally, execute these operations last, from left to right.

For example, in the expression \( \frac{-2 - 4^2}{3(-6)} \), the exponent is evaluated first, simplifying \( 4^2 \) to \( 16 \). Then, subtraction follows in the numerator, and multiplication in the denominator, ultimately leading to the division of the simplified numerator and denominator. By adhering to the order of operations, you achieve accuracy and maintain a universal approach to solving expressions.