Question

For Exercises \(1-74,\) simplify. $$39 \div 3+(24-30)-5^{2}+[-21-(-13)]$$

Short answer

Answer: -26

Step-by-step solution

Simplify the division and exponentiation

The first step is to perform any division and exponentiation operations in the expression. $$39 \div 3 = 13$$ $$5^{2} = 25$$ So the expression becomes: $$13 + (24 - 30) - 25 + [-21 - (-13)]$$

Simplify the parentheses

Now we need to simplify the expressions inside the parentheses: $$(24 - 30) = -6$$ $$[-21 - (-13)] = -21 + 13 = -8$$ So the expression becomes: $$13 - 6 - 25 - 8$$

Simplify addition and subtraction operations

Finally, we perform addition and subtraction operations from left to right: $$13 - 6 = 7$$ $$7 - 25 = -18$$ $$-18 - 8 = -26$$ Therefore, the simplified expression is: $$\boxed{-26}$$

Key concepts

Order of Operations

Understanding the order of operations is crucial when simplifying mathematical expressions. This set of rules, often memorized by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), tells us the sequence in which we should perform operations to correctly solve an expression. Always start by solving operations inside parentheses, then move on to exponents. After that, tackle any multiplication or division from left to right, and finally, manage addition and subtraction from left to right. Failing to follow these steps can lead to incorrect answers, as changing the order can significantly change the result of the calculation.
When we look at our exercise consisting of several operations, using PEMDAS ensures that we first divide \(39 \div 3\) and deal with the exponent in \(5^{2}\), before addressing any operations in parentheses or the final addition and subtraction.

Exponentiation

Exponentiation is an operation that raises a number to the power of another number. It's the repeated multiplication of a number by itself, and it's a key concept when simplifying mathematical expressions. In our exercise, \(5^{2}\) represents 5 multiplied by itself, leading to 25. It's essential to solve any exponents right after any parentheses but before performing any multiplication, division, addition, or subtraction that are not enclosed in parentheses. This operation has higher precedence than most other arithmetical operations, which is why it's crucial to solve it early in the simplifying process.

Arithmetic Operations

Arithmetic operations include addition, subtraction, multiplication, and division. When simplifying mathematical expressions, it's important to execute these operations in the correct order after parentheses and exponentiation. Multiplication and division are of equal precedence, so you work from left to right. The same applies to addition and subtraction, which are also of equal precedence to each other and are performed after multiplication and division. In practice, this means that once we've dealt with any parentheses or exponents, we then perform any divisions or multiplications as we encounter them when we read from left to right, followed by the same process for addition and subtraction. For instance, in the final steps of our exercise, we handle the subtraction operations sequenced from the leftmost part of the expression to the right, resulting in \(-26\).

Parentheses in Math

Parentheses in math signify that the operations contained within them should be performed first before any outside operations are taken into account, in accordance with the order of operations. They can also indicate grouping for clarification, which is crucial for maintaining the intended arithmetic structure. In the given example, we simplify expressions inside the parentheses such as \((24 - 30)\) and \([-21 - (-13)]\) before moving on to other operations. Moreover, the presence of brackets around negative numbers or expressions, like \([-21 - (-13)]\), signals that we should consider the subtraction of a negative number as addition. By addressing the contents within parentheses at the start, it simplifies the process and allows for a smoother operation flow, ultimately resulting in clearer and more accurate simplification.