Problem 74 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. Then simplify the answer. $$ \frac{2 \cdot 4^{2}}{1+3^{2}-2} $$

Step-by-Step Solution

Answer

The simplified answer is 4.

Step 1: Identify Operands And Apply BIDMAS/BODMAS Rule

Our mathematical expression does not have any brackets, so we first deal with indices. \(4^2\) means 4 is being multiplied by itself twice and \(3^2\) means 3 is being multiplied by itself twice. Thus, we calculate these first.

Step 2: Calculate Exponentiation

Calculating \(4^2= 16\) and \(3^2= 9\). So, our expression now becomes \(\frac{2 \cdot 16}{1+9-2}\)

Step 3: Perform Multiplication

Now, dealing with multiplication, we have \(2 \cdot 16 = 32\). So, our expression now becomes \(\frac{32}{1+9-2}\)

Step 4: Perform Addition And Subtraction

Now, deal with addition and subtraction. We perform \(1+9-2 = 8\). So, our expression now becomes \(\frac{32}{8}\)

Step 5: Perform Division

Finally, we perform division. \(32 \div 8 = 4\)

Key Concepts

BIDMAS/BODMAS

BIDMAS (or BODMAS) is an acronym that stands for the order of operations we should follow when solving mathematical expressions.

• B - Brackets
• I/D - Indices (or O for Orders which include powers and roots)
• D/M - Division and Multiplication (from left to right)
• A/S - Addition and Subtraction (from left to right)

Understanding and applying BIDMAS/BODMAS is crucial to obtaining the correct answer when simplifying expressions. It ensures that we consistently apply the operations in the right sequence. For instance, exponentiation is performed before multiplication and division. Always work through expressions using this order to avoid mistakes.

exponentiation

Exponentiation involves raising a number, called the base, to the power of another number. It is indicated by the superscript numeral. In \[ a^n \], \(a\) is the base, and \(n\) is the exponent.

In our original exercise, we have \[ 4^2 \] which equals \[ 4 \times 4 = 16 \], and \[ 3^2 \] which equals \[ 3 \times 3 = 9 \].

Exponentiation is a fundamental operation in mathematics that allows us to represent repeated multiplication efficiently. Always remember that \[ a^n \] means multiplying \(a\) by itself \(n\) times. In the context of BIDMAS/BODMAS, it must be completed as early as possible in complex calculations.

simplification

Simplification in mathematics involves reducing an expression to its simplest form, which often makes it easier to understand and solve.
To simplify the given expression \[ \frac{2 \times 16}{1+9-2} \], you perform operations step-by-step applying BIDMAS/BODMAS.

After completing exponentiation, we handle multiplication: \[ 2 \times 16 = 32 \]. Then, we simplify the expression by performing addition and subtraction in the denominator: \[ 1+9-2 = 8 \].

Simplification makes calculations less complex and results straightforward, keeping only what's necessary in an expression for clarity and further calculation, as seen in the resulting fraction \[ \frac{32}{8} \].

evaluation

Evaluation in mathematics refers to the process of calculating the exact value of an expression. Once the expression is simplified, as we have already in \[ \frac{32}{8} \], we evaluate it through division, resulting in \[ 4 \].

In the context of the given exercise, evaluation finalizes the problem-solving process, providing a concrete numerical answer. By breaking down complex expressions step-by-step using BIDMAS/BODMAS and performing the operations in sequence, students arrive at the accurate result. This process underscores the importance of both understanding each mathematical operation and the order in which they should be applied.