Question

For Exercises \(1-74,\) simplify. $$(-2) \cdot(5-8)^{2} \div 6+(4-2)$$

Short answer

Question: Simplify the following expression according to the order of operations (PEMDAS): $$(-2) \cdot(5-8)^{2} \div 6+(4-2)$$ Answer: After simplifying the expression according to the order of operations, we get the result $$-1$$.

Step-by-step solution

Evaluate expressions inside parentheses

First, we will evaluate the expressions inside the parentheses before performing any other operations. $$(-2) \cdot(5-8)^{2} \div 6+(4-2)$$ The first parentheses: \((5-8) = -3\) The second parentheses: \((4-2) = 2\) Now we have: $$(-2) \cdot(-3)^{2} \div 6+2$$

Apply the exponent

Next, we will apply the exponent to the result inside the parentheses from Step 1. $$(-2) \cdot(-3)^{2} \div 6+2$$ The exponent: \((-3)^{2} = 9\) Now we have: $$(-2) \cdot 9 \div 6+2$$

Perform multiplication and division from left to right

Now we will multiply and divide from left to right as per the order of operations. $$(-2) \cdot 9 \div 6+2$$ First, we perform the multiplication: \((-2) \cdot 9 = -18\) Now we have: $$-18 \div 6+2$$ Next, we perform the division: \(-18 \div 6 = -3\) Now we have: $$-3+2$$

Perform addition

Finally, we will perform the addition operation to get the simplified result. $$-3+2$$ The addition result: \(-3 + 2 = -1\) So, the simplified expression is: $$-1$$

Key concepts

Order of Operations

Understanding the order of operations, often abbreviated as PEMDAS or BODMAS, is crucial when simplifying algebraic expressions. It helps determine the sequence in which you should perform mathematical operations to get the correct answer. The acronym stands for Parentheses (or Brackets), Exponents (or Orders), Multiplication and Division, and Addition and Subtraction.

To apply the order of operations in our example, we start by simplifying the terms within parentheses. After parentheses, exponents come next, followed by multiplication and division, completed from left to right. Lastly, we handle addition and subtraction, again working from left to right. If you forget this sequence, you might miscalculate your expression, leading to an incorrect solution. Keeping the order of operations in mind can transform a complex problem into an easy-to-follow set of smaller steps.

Exponentiation

Exponentiation is a form of repeated multiplication where a number, known as the base, is multiplied by itself a certain number of times specified by the exponent. For instance, in the expression \( (-3)^2 \), the base is -3, and the exponent is 2, meaning that -3 is multiplied by itself once (\( -3 \times -3 \)).

Exponents have their own unique properties, such as any number raised to the power of 2 (squared) will result in a positive value if the base is a real number. That is why in our example, \( (-3)^2 \) becomes 9 and not -9. Whenever you encounter exponents in expressions, it's important to apply them before moving on to multiplication, division, addition, or subtraction, unless the expression is within parentheses.

Arithmetic Operations

Arithmetic operations are the building blocks of mathematics. They include addition, subtraction, multiplication, and division. When simplifying expressions, understanding how to correctly apply these operations is vital.

In the given problem, once we've dealt with parentheses and exponents, we move on to multiplication and division. The expression \( (-2) \cdot 9 \) instructs us to multiply -2 by 9 before dividing the result by 6. Then, we address the addition, adding the quotient to 2. Without properly applying these arithmetic operations within the framework of the order of operations, it's easy to arrive at an inaccurate answer. A precise approach to using arithmetic operations in the right order ensures we simplify the algebraic expressions correctly.