Problem 90 - Beginning and Intermediate Algebra

Question

Simplify. $$ \frac{3^{2}+4}{5} $$

Step-by-Step Solution

Answer

The simplified expression is \(2.6\) or \(\frac{13}{5}\).

Step 1: Calculate the Exponential Term

First, calculate the expression inside the exponent. We have \(3^2\). This means multiplying 3 by itself: \(3 \times 3 = 9\).

Step 2: Simplify the Expression Inside the Numerator

Add the result from Step 1 to the remaining term in the numerator. So, \(9 + 4 = 13\).

Step 3: Perform the Division

Divide the result from Step 2 by the denominator. That is \(\frac{13}{5}\), which simplifies to \(2.6\) or can be left as a fraction if needed.

Key Concepts

Exponentiation

Exponentiation is a mathematical operation that involves raising a number to a power. The number that is being multiplied is called the base, and the power describes how many times to multiply the base by itself. When you see something like \(3^2\), the base is 3, and the exponent, or power, is 2. This means you multiply 3 by itself once, resulting in \(3 \times 3\). For our given example, this calculates to 9. Understanding exponentiation is crucial because it simplifies expressions and makes them easier to work with, especially in equations that involve larger numbers. Keeping track of what each part of the expression represents can help avoid mistakes as you work through a problem.

Numerator and Denominator

In fractions, the terms **numerator** and **denominator** are very important. The numerator is the top part of a fraction, and it represents how many parts of the whole you have. The denominator is the bottom part, showing into how many parts the whole is divided.For example, in the fraction \(\frac{3^2 + 4}{5}\), \(3^2 + 4\) is the numerator, and 5 is the denominator. Understanding what each part of the fraction represents helps when simplifying or calculating fractions. By focusing on simplifying the numerator first, complex fractions become much easier to tackle. In the given problem, we started by simplifying the numerator with the exponentiation and then adding up the needed parts.

Fraction Division

Fraction division involves dividing the numerator by the denominator. This operation requires simplifying the numerator and denominator to the most straightforward terms possible before performing the division.In the exercise, after calculating \(3^2 + 4\) to get 13, we then divide 13 by 5. You could end with a decimal result, 2.6, or keep it in fractional form, \(\frac{13}{5}\). Both forms are valid, but choosing one depends on the context required by your math problem.Being able to move seamlessly between a fraction and its decimal form is important in mathematics, as certain problems might call for one format over the other. Simplifying both the numerator and denominator before dividing is a key step in making sure fraction division is handled correctly.