Problem 35 - Algebra 1: Concepts and Skills

Question

Evaluate the variable expression when a = 3 and c = 5. $$ \left(a^{2}\right) \cdot c $$

Step-by-Step Solution

Answer

The evaluated value of the expression is 45.

Step 1: Substitute The Variables

Replace each variable in the expression with the value provided. This gives \( (3^{2}) \cdot 5 \).

Step 2: Solve The Exponential Part

Calculate the value of \( 3^{2} \) first due to the order of operation (BIDMAS/BODMAS). This gives \( 9 \cdot 5 \).

Step 3: Solve The Multiplication

Calculate \( 9 \cdot 5 \) to get the final result. The answer is 45.

Key Concepts

Substitution

Substitution is the process of replacing variables in an expression with their given values. Think of variables as placeholders representing numbers, and each variable can be substituted by its assigned number.

Consider the expression \((a^{2}) \cdot c\). To evaluate it, first, identify the values given for the variables, which are \(a = 3\) and \(c = 5\). Replace \(a\) with 3 and \(c\) with 5 in the expression. This transforms the expression into \((3^{2}) \cdot 5\).

This technique helps in simplifying complex algebraic expressions and finding numerical answers quickly.

Order of Operations

The order of operations is a set of rules that dictate the correct sequence to evaluate mathematical expressions. Commonly known by the acronym BIDMAS or BODMAS, which stands for:

• Brackets
• Indices (or Order for exponents)
• Division and Multiplication (from left to right)
• Addition and Subtraction (from left to right)

In the expression \((3^{2}) \cdot 5\), you first solve the exponent \((3^{2})\) because exponents come before multiplication in the order of operations.

Always follow these rules to avoid mistakes in calculations, especially in complex expressions with multiple operations.

Exponents

Exponents denote repeated multiplication of a number by itself. In mathematical terms, when you see \(a^{2}\), it means \(a \times a\).

For example, in the expression \((3^{2})\), the exponent 2 tells you to multiply 3 by itself. Thus, you calculate \(3 \times 3\), which equals 9. This simplifies our expression to \(9 \cdot 5\).

Understanding how to calculate exponents is essential for solving expressions involving powers, making it a crucial skill in algebra.