Problem 15 - Prealgebra and Introductory Algebra

Question

Simplify. $$(-1) \cdot(4-7)^{2} \div 9+6-3-4(2)$$

Step-by-Step Solution

Answer

The simplified result of the expression is -6.

Step 1: Simplify Within Brackets

Begin by simplifying the operation within brackets. Here it is \(4-7 = -3\). So, the expression becomes \( (-1) \cdot (-3)^{2} \div 9 + 6 - 3 - 4(2) \)

Step 2: Carry Out Exponentiation

Procees with the exponentiation to find \((-3)^2 = 9\). So, the expression now becomes \((-1) \cdot 9 \div 9 + 6 - 3 - 4(2)\)

Step 3: Carry Out Multiplication and Division From Left to Right

Next, carry out multiplication and division operations from left to right. Start with multiplication: \((-1) \cdot 9 = -9\). So, the expression now becomes \(-9 \div 9 + 6 - 3 - 4(2)\). Then proceed with division: \(-9 \div 9 = -1\). So, the expression now becomes \(-1 + 6 - 3 - 4(2)\)

Step 4: Carry Out Addition and Subtraction From Left to Right

Now, carry out addition and subtraction operations from left to right. Begin with addition: \(-1 + 6 = 5\). So, the expression now becomes \(5 - 3 - 4(2)\), then proceed with subtraction: \(5 - 3 = 2\). So, the expression now becomes \(2 - 4 \cdot 2 = 2 - 8\)

Step 5: Finish Operation

Now, just subtract \(2 - 8 = -6\).

Key Concepts

Simplification

Simplification is the process of making complex expressions easier to understand by reducing them to their simplest form.
To begin simplifying an expression, always start with the operations inside brackets or parentheses.
In our problem,

• First, we calculate the expression inside the brackets: \(4 - 7 = -3\).

Breaking down complex expressions first helps to prevent mistakes and makes subsequent operations clearer.

Exponentiation

Exponentiation involves raising a number to a power, which is another term for repeated multiplication.
Our expression requires us to calculate \((-3)^2\), meaning we need to multiply \(-3\) by itself.

• This results in: \((-3) imes (-3) = 9\).

Notice that squaring a negative number yields a positive product. This step clears the way for more straightforward arithmetic operations that follow.

Multiplication and Division

The order of operations dictates that multiplication and division should be addressed after exponents but before addition and subtraction.
This sequence is handled from left to right.

• Start by multiplying: \((-1) imes 9 = -9\).
• Then move to division: \(-9 \div 9 = -1\).

Only once these operations are complete can we move forward to tackling addition and subtraction, continuing from left to right.

Addition and Subtraction

Addition and subtraction come after exponentiation, multiplication, and division under the order of operations.
To solve, we calculate sequentially from left to right.

• Add: \(-1 + 6 = 5\).
• Next, subtract: \(5 - 3 = 2\).
• And, finally, subtract the multiplication: \(2 - 4(2) = 2 - 8 = -6\).

This step-by-step approach ensures accuracy and results in the final answer of the expression.