Problem 96 - Beginning and Intermediate Algebra

Question

Simplify. $$ \frac{-2-4^{2}}{3(-6)} $$

Step-by-Step Solution

Answer

The simplified expression is 1.

Step 1: Evaluate Powers

First, evaluate the power expression in the numerator. The term is \( 4^2 \). Calculate \( 4^2 \), which equals 16.

Step 2: Simplify the Numerator

Substitute the result from Step 1 into the numerator: \( -2 - 4^2 = -2 - 16 \). Now, perform the subtraction to get \( -18 \).

Step 3: Simplify the Denominator

Look at the denominator, which is \( 3(-6) \). Multiply the terms: \( 3 \times -6 = -18 \).

Step 4: Divide Numerator by Denominator

Now divide the simplified numerator by the simplified denominator: \( \frac{-18}{-18} \). Simplify the fraction to get 1.

Key Concepts

Exponentiation

Exponentiation is a mathematical operation where a number, called the base, is multiplied by itself a specific number of times, indicated by the exponent. In our exercise, the base is 4, and the exponent is 2, which we write as \( 4^2 \). This tells us to multiply 4 by itself once: \( 4 \times 4 \).

• The base is the number that gets multiplied.
• The exponent is the number of times the base is multiplied by itself.

When we calculate \( 4^2 \), we find that it equals 16. Exponentiation is a shortcut to repeated multiplication, making it easier to work with large numbers.
Understanding exponentiation is crucial for solving algebra problems efficiently.

Numerator and Denominator

In a fraction, the numerator and denominator are the two main components. The numerator is the number above the fraction line and indicates how many parts we have. The denominator is the number below the line and shows into how many equal parts the number is divided.In our exercise, the fraction is \( \frac{-2-4^2}{3(-6)} \).
- The numerator is \(-2 - 16\), which simplifies to \(-18\).- The denominator is \(3(-6)\), simplifying to \(-18\).Understanding these components is essential, as they determine the value of the fraction when divided. The numerator and denominator operations follow the same rules of arithmetic, implying subtraction, addition, multiplication, and so on, depending on the problem.

Fraction Simplification

Fraction simplification involves reducing a fraction to its simplest form where the numerator and denominator have no common factors other than 1. In our problem, after simplifying the numerator to \(-18\) and the denominator to \(-18\), we create the simple fraction \( \frac{-18}{-18} \). To simplify, you divide the numerator by the denominator. - Since \(-18\) divided by \(-18\) equals 1, the simplified fraction is simply 1.During simplification, remember to:

• Identify common factors in the numerator and denominator.
• Divide both by the greatest common factor.
• Acknowledge that dividing a number by itself equals 1.

Learning to simplify fractions is a key skill in algebra, ensuring calculations are manageable and results more understandable.