Problem 29 - Algebra A Combined Function

Question

Simplify each expression. $$ 5 \cdot 3^{2} $$

Step-by-Step Solution

Answer

The simplified expression is 45.

Step 1: Evaluate the Exponent

First, calculate the exponent in the expression. The expression inside the exponent is \(3^2\), which means \(3\) multiplied by itself. So, \(3^2 = 3 \times 3 = 9\).

Step 2: Multiply by the Coefficient

Now that we have simplified the exponent to \(9\), we need to multiply this result by the coefficient \(5\) in front of the expression. Perform the multiplication: \(5 \times 9 = 45\).

Step 3: Final Simplified Result

After performing the multiplication, the final simplified expression is \(45\).

Key Concepts

Simplifying Expressions

Simplifying expressions is a fundamental skill in mathematics that involves reducing complex expressions to their simplest form. This process often involves removing parentheses, combining like terms, and performing arithmetic operations.

In the given example, "Simplify each expression", the goal is to take the original expression, which includes an exponent and a multiplication, and transform it into its simplest numerical value. Simplifying expressions makes calculations easier and results more understandable.

To effectively simplify an expression:

• Identify and solve exponents (powers) first; they tell you how many times to multiply the base number by itself.
• Carry out multiplications or any other arithmetic operations as per the order of operations.
• Check the expression to ensure it's fully simplified, meaning that no further calculations can be performed.

In our case, starting with the exponent and followed by multiplication gave us a clean and clear result of the expression as 45.

Order of Operations

The order of operations is a rule that defines the sequence in which different parts of a mathematical expression should be solved. This rule, often remembered by the acronym PEMDAS, stands for:

• P - Parentheses
• E - Exponents
• M - Multiplication
• D - Division
• A - Addition
• S - Subtraction

Following this order is crucial for obtaining the correct result. In mathematical expressions, operations must be performed in this specific sequence to ensure accurate outcomes.

In our expression, the order of operations plays a vital role:

• First, solve the exponent, which is 3 squared (or \(3^2\)), resulting in 9. Remember, exponents are second in the PEMDAS order right after any parentheses.
• Next, multiplication is performed: \(5 \times 9\).Finalizing this gives us the simplified value of 45.

By following this order, computations become standardized, preventing any misinterpretation and ensuring consistency across different problems.

Multiplication

Multiplication is one of the four basic arithmetic operations, representing repeated addition. It simplifies the process of adding a number to itself multiple times.

In our example, multiplication occurs in two distinct phases:

• Within the exponent, where 3 is multiplied by itself (since \(3^2 = 3 \times 3 = 9\)).
• Between the coefficient 5 and the result from the exponent, giving us \(5 \times 9\).

This final multiplication step results in the simplified expression, which is 45.

It's essential to understand that multiplication is a straightforward operation that can drastically reduce the time needed for solving math problems, especially when dealing with exponents or larger numbers. Consistent practice in executing multiplication smoothly and accurately can enhance overall math skills.