Problem 32 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. \(2 \cdot 3^{2}-7\)

Step-by-Step Solution

Answer

The evaluation of the expression \(2 \cdot 3^{2}-7\) is 11.

Step 1: Evaluate the Exponent (Indices) First

In the given problem, the expression 3^{2} is an exponent (or power). So, we'll calculate that first. The result will be: \(2 \cdot 3^{2}-7 = 2 \cdot 9 -7\).

Step 2: Perform Multiplication

Following BIDMAS or PEMDAS, multiplication should be done next. The expression now becomes: \(2 \cdot 9 -7 = 18 -7\).

Step 3: Execute Subtraction

Lastly, execute the subtraction operation to find the result of the entire expression. The expression now becomes: \(18-7 = 11\).

Key Concepts

Understanding Exponents

Exponents are a mathematical way to express repeated multiplication of a number by itself. In our exercise, we came across the expression \(3^2\). This is a simple exponent where the base is 3 and the exponent is 2. The exponent indicates that the number 3 should be multiplied by itself once, resulting in \(3 \times 3 = 9\). Exponents simplify expressions.

• The base: the number that is being multiplied, which is 3 in this case.
• The exponent: the number of times the base is used in the multiplication, which is 2 here.

By calculating the exponent first, we simplify the expression and make it easier to manage in later steps. Understanding how exponents work is crucial in both basic and advanced math problems.

The Role of Multiplication

Once the exponent is calculated, the next step in the order of operations is to tackle multiplication. In the problem, after simplifying the exponent, we have the term \(2 \cdot 9\). This is a straightforward multiplication problem.

• Multiply the two numbers as indicated.
• In this instance, \(2 \cdot 9 = 18\).
• Multiplication helps in scaling numbers, as shown here by scaling 9 by a factor of 2.

By completing the multiplication in the right sequence, we simplify the subsequent operations and keep mistakes at bay.

Executing Subtraction

With the exponents and multiplication out of the way, we now perform subtraction, which is usually one of the last operations after simplifying an expression per the BIDMAS/PEDMAS rules. We arrive at \(18 - 7\) from the prior steps.

• Subtract the second number from the first (\(18 - 7\)).
• Here, you simply find what remains when seven is subtracted from eighteen, resulting in 11.

Subtraction helps to decrease the value according to the problem's requirements. Ensure subtraction is executed after other higher-priority operations to maintain accuracy.