Problem 81 - Algebra 1: Concepts and Skills

Question

NUMERICAL EXPRESSIONS Evaluate the expression. Then simplify the answer. (Lesson 1.3) $$ \frac{4^{2}+2}{2} $$

Step-by-Step Solution

Answer

The final simplified answer is 9.

Step 1: Calculate the value in the numerator

Firstly, deal with the exponent in the numerator. \(4^{2}\) equals 16. Then, add 2 to this value to get \(16 + 2 = 18\). The numerator is 18.

Step 2: Divide by denominator

Since there are no operations left to perform other than division, divide the numerator by the denominator. That is \( \frac{18}{2} = 9 \).

Key Concepts

Exponentiation

Exponentiation is the process of raising a number to the power of another number. In the expression \(4^2\), the base is 4 and the exponent is 2. This means you multiply 4 by itself, calculating \(4 \times 4\) which equals 16.
It's important to understand that the exponent shows how many times the base is used as a factor.
Exponents can simplify multiplication expressions, turning repeated multiplication into a more compact form:

• For example, \(2^3 = 2 \times 2 \times 2 = 8\).
• Zero exponent, \(a^0\), equals 1 for any non-zero base \(a\).
• Negative exponents represent fractions: \(a^{-n} = \frac{1}{a^n}\).

Exponents are a crucial part of the "order of operations".

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 the exercise, the fraction \(\frac{18}{2}\) is simplified by division.
Here, 18 is divided by 2, resulting in 9, which means:\[\frac{18}{2} = 9\]
Simplification helps in getting fractions to their simplest form:

• Divide both the numerator and denominator by their greatest common divisor (GCD).
• Keep simplifying until no further division is possible.
• Sometimes this step follows evaluating expressions involving fractions.

Simplifying fractions makes them easier to work with, and helps understand the number better.

Order of operations

The order of operations determines the sequence in which mathematical operations are performed to ensure consistent results. We follow the BODMAS/BIDMAS rule:
- Brackets
- Orders (Exponents and roots)
- Division and Multiplication (left to right)
- Addition and Subtraction (left to right)
In the given exercise, the numerator is calculated first by evaluating the exponent \(4^2\), which simplifies to 16.
Subsequently, you add 2, resulting in 18 before dividing by 2.
Following the order of operations ensures each expression is simplified correctly and prevents mistakes in calculations:

• Always start solving from the innermost brackets or parentheses.
• Solve any exponents before any multiplication or division operations.
• Look for any addition or subtraction last.

Using proper order keeps expressions valid and calculations accurate.