Question

For Exercises \(1-74,\) simplify. $$\left.\left.\left\\{(-3)^{2}+4\right](8-20) \div(-2)\right]\right\\}+(-5) \sqrt{25 \cdot 4}$$

Short answer

Question: Simplify the given expression: $$\left.\left.\left\\{(-3)^{2}+4(8-20)\div (-2)\right]\right\\}+\sqrt{25\cdot4}(-5)$$ Answer: The simplified expression is 28.

Step-by-step solution

Evaluate Powers and Roots

First, we need to evaluate any powers and roots present in the expression. In our case, there is only one power term "(-3)^2" and one root term "sqrt(25 * 4)". $$(-3)^{2}=9$$ $$\sqrt{25 \cdot 4}=\sqrt{100}=10$$ Now the expression becomes: $$\left.\left.\left\\{(9+4\right](8-20) \div(-2)\right]\right\\}+(-5)(10)$$

Perform Multiplication and Division

Now, we need to do the multiplication and division operations from left to right: $$9 + 4 = 13$$ $$8 - 20 = -12$$ $$(-5)(10) = -50$$ $$-12 \div (-2) = 6$$ So the expression becomes: $$\left.\left.\left\\{(13)(6)\right]\right\\}+(-50)$$

Perform Addition and Subtraction

Finally, we need to do the addition and subtraction operations from left to right: $$13 \cdot 6 = 78$$ Now the expression becomes: $$78 + (-50) = 78 - 50 = 28$$ So, the simplified expression is: $$\boxed{28}$$

Key concepts

Order of Operations

Understanding the order of operations is fundamental in simplifying mathematical expressions. It's a rule that tells us the sequence in which we should solve different parts of a mathematical problem. This sequence is often remembered by the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

In the given exercise, the order of operations is applied step by step. Starting with evaluating powers, then roots, followed by multiplication and division, and finally, addition and subtraction. The correct application of these rules ensures that no matter who solves the problem, they will arrive at the same answer if done correctly. By adhering strictly to this sequence, students avoid common mistakes and can simplify complex expressions down to a single value effectively.

Evaluating Powers and Roots

Evaluating powers and roots is usually handled early on when simplifying expressions, right after solving anything within parentheses. Powers refer to the number of times a number is multiplied by itself, such as in \( (-3)^2 \) where -3 is multiplied by itself to give 9. The roots, on the other hand, are the inverse operation of powers, represented by the radical symbol \( \sqrt{} \). For example, \( \sqrt{25 \cdot 4} \) asks for a number, which when multiplied by itself, gives the product of 25 and 4, which is 100. The square root of 100 is 10.

In the textbook exercise, recognizing that these operations must be performed prior to any other operations is key. Mistaking these steps can lead to incorrect results. By evaluating powers and roots first, students lay the groundwork for the rest of the simplification process, as demonstrated in the solution. It is important to remember that when dealing with powers, the sign of the base and the power itself can significantly affect the outcome.

Arithmetic Operations

Arithmetic operations include addition, subtraction, multiplication, and division. These are the building blocks of most mathematical computations. After evaluating powers and roots, the expression is further simplified by performing the arithmetic operations in the correct order.

In the example provided, after handling the powers and roots, the solver correctly moves to multiplication and division before dealing with addition and subtraction. One helpful way to avoid errors is to rewrite the expression after each step. This step-by-step simplification helps clarify which operations remain, as intermediate steps are often where students can get lost.

For instance, \( 8 - 20 \) yields -12, and \( -5 \times 10 \) gives -50. When these intermediate steps are handled with care, the final addition or subtraction, as in \( 78 - 50 \) to get 28, falls into place easily. By patiently working through each arithmetic operation, ensuring that multiplication and division take precedence over addition and subtraction, students can confidently simplify expressions to their most elementary form.