Problem 2 - College Algebra and Calculus: An Applied Approach

Question

Evaluate the expression. Write fractional answers in simplest form.\(3 \cdot 3^{5}\)

Step-by-Step Solution

Answer

The simplest form of the expression \(3 \cdot 3^{5}\) is 729.

Step 1: Calculate the Exponent

Use the rule of exponentiation for the calculation of \(3^5\). This means multiplying 3 by itself five times.

Step 2: Multiply the Result by 3

The next step is to multiply the resulting value from Step 1 with 3 to get the final answer.

Key Concepts

Simplifying Fractions

Simplifying fractions is the process of making a fraction as simple as possible. It's important in algebra to represent values in their simplest form to make calculations and comparisons easier. A fraction's simplified form has no common factors between the numerator and the denominator other than 1. For example, if we have a fraction like \(\frac{8}{12}\), we can simplify it:

• Find the greatest common divisor (GCD) of 8 and 12, which is 4.
• Divide both the numerator and the denominator by their GCD: \(\frac{8}{12} = \frac{8 \div 4}{12 \div 4} = \frac{2}{3}\).

In our exercise, simplifying might not seem directly applicable since we mainly deal with the multiplication of whole numbers, but understanding simplification will always be useful for when the result of multiplication yields a fraction that can be reduced.

Multiplication

Multiplication is a fundamental arithmetic operation used to calculate the total of adding a number to itself a specific number of times. It is often indicated by the "\(\cdot\)" symbol or simply by juxtaposition in algebra.
In the provided exercise, we need to multiply 3 by the result of \(3^5\).
Here's how we approach it:

• Calculate \(3^5\) first. This means multiplying 3 by itself five times: \(3 \times 3 \times 3 \times 3 \times 3 = 243\).
• Next, multiply the result by 3: \(3 \times 243 = 729\).

Understanding multiplication is crucial, whether we're dealing with simple numbers or more complex algebraic expressions.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They may contain terms with exponents, such as the expression \(3 \cdot 3^{5}\) seen in the exercise.
Here are the basic components:

• Numbers: These are the constants like 3 in our exercise.
• Variables: Often represented by letters, they stand for unknown or changeable values.
• Operations: Include addition, subtraction, multiplication, division, and exponentiation (like \(3^5\)).

When evaluating algebraic expressions, it's important to follow the order of operations: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (also from left to right). In our case, the power (exponentiation) is done before multiplication.