Problem 50 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. (Lessons 1.2,1.3) $$ 5 \cdot 2+4^{2} $$

Step-by-Step Solution

Answer

The evaluated result of the expression \(5 \cdot 2+4^{2}\) is 26

Step 1: Identify Operations

The expression to be evaluated is \(5 \cdot 2+4^{2}\). The operations involved in this expression are multiplication, addition and exponent

Step 2: Apply Exponents

According to BIDMAS/PEMDAS, exponents are dealt with second, following brackets. We have one exponent in the expression, \(4^{2}\), which equals \(4 \cdot 4\) or 16.

Step 3: Perform multiplication

Next, according to BIDMAS/PEMDAS, we perform multiplication operation. In the expression, we have \(5 \cdot 2\), which equals 10.

Step 4: Perform Addition

Finally, we perform the addition. According to BIDMAS/PEMDAS, addition is done after multiplication and division. So add the results from Step 2 and Step 3 together. So, \(10 (from Step 3) + 16 (from Step 2) = 26\).

Key Concepts

PEMDAS

PEMDAS is a helpful acronym used to remember the order of operations in mathematics. It stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). This order ensures that everyone performs calculations consistently, leading to the correct answer.

• **Parentheses**: First, solve everything within parentheses or brackets.
• **Exponents**: Second, deal with any exponents or powers, such as squares or cubes.
• **Multiplication and Division**: Third, perform these operations as you encounter them from left to right.
• **Addition and Subtraction**: Lastly, these are also solved from left to right.

By following PEMDAS, we are certain to solve expressions in a structured and systematic way, avoiding errors in calculations.

BIDMAS

BIDMAS is another acronym similar to PEMDAS but commonly used in the UK and other countries. It stands for Brackets, Indices, Division and Multiplication, Addition and Subtraction. The concept is very similar, just with different terms used for some steps.

• **Brackets**: Solve expressions inside brackets first.
• **Indices**: These are known as exponents in PEMDAS. Deal with them next.
• **Division and Multiplication**: Perform these operations as they appear from left to right.
• **Addition and Subtraction**: Finally, handle these operations from left to right.

The fundamental idea is the same: follow a particular sequence to solve expressions accurately. Whether you use BIDMAS or PEMDAS, the sequence remains intuitive and helps keep math fun and error-free.

Exponents

Exponents, also called powers, represent how many times a number, known as the base, is multiplied by itself. For example, in the expression \(4^2\), 4 is the base and 2 is the exponent, meaning \(4 \cdot 4\).
Understanding exponents is crucial because they come directly after parentheses or brackets in both PEMDAS and BIDMAS.
Using exponents efficiently simplifies math expressions and solves equations faster, especially when dealing with large numbers.

• **Identifying Exponents**: Before any other operation, check for and compute exponents in the expression.
• **Calculation**: In our expression, \(4^2 = 16\). This is an essential step for solving the equation correctly.

Exponents transform complex multiplications into concise terms, making calculations more straightforward.

Multiplication

Multiplication is one of the core arithmetic operations. It is repeated addition. For instance, \(5 \cdot 2\) indicates that 5 is added to itself 2 times (or vice versa), equaling 10.
When applying PEMDAS or BIDMAS, multiplication is handled right after exponents and before addition and subtraction. This ensures that complex expressions are broken down logically and sequentially.

• **Simplifying Multiplication**: Handle all multiplication in your expression following the order of operations.
• **Example in Practice**: In the expression \(5 \cdot 2\), the result is 10, which is combined with other computed values at a later stage.

Multiplication can dramatically reduce the number of steps in solving equations and makes calculations efficient.

Addition

Addition is among the most basic mathematical operations, combining numbers into their sum. In the order of operations, it is performed after multiplication and division are complete.
For the given expression, once we have computed the exponents and performed the multiplication, addition is the final step. Following the proper sequence ensures that you don't mix up operations, leading to incorrect results.

• **Order Matters**: Always perform addition after solving multiplication and division.
• **Application Example**: Once you have \(10\) from multiplication and \(16\) from exponents, simply add them to get \(26\).

Even though addition seems straightforward, adhering to the proper order is critical to solving complex expressions accurately.