Question

For Exercises \(1-74,\) simplify. $$\\{19-2[4+(-9)]\\}-18 \div \sqrt{25-16}$$

Short answer

Question: Simplify the given expression: \\{19-2[4+(-9)]\\}-18 ÷ √(25-16) Answer: 23

Step-by-step solution

Evaluate the expression inside the brackets

First, focus on the expression inside the brackets: \([4+(-9)]\). Add the numbers: \(4+(-9) = -5\). The expression becomes: \\{19-2(-5)\\}-18 ÷ √(25-16).

Simplify the expression inside the curly braces

We need to perform the multiplication operation next: \(19-2(-5) = 19 + 10 = 29\). The expression becomes: \\{29\\}-18 ÷ √(25-16).

Evaluate the expression inside the square root

Now, we need to focus on the expression inside the square root: \((25-16)\). Calculate the difference: \(25-16 = 9\). The expression becomes: \\{29\\}-18 ÷ √(9).

Evaluate the square root

Next, find the square root of the number inside the square root: √(9) = 3. The expression becomes: \\{29\\}-18 ÷ 3.

Perform the division operation

Now, let's divide 18 by 3: \(18 ÷ 3 = 6\). The expression becomes: \\{29\\}-6.

Perform the subtraction operation

Lastly, subtract 6 from 29: \(29 - 6 = 23\). The simplified expression is: 23.

Key concepts

Order of Operations

Understanding the order of operations is crucial for simplifying expressions accurately. This ensures that calculations yield the correct result. Remember the acronym PEMDAS to guide you:

• Parentheses - Always solve expressions inside parentheses first.
• Exponents - Calculate powers and roots next.
• Multiplication and Division - From left to right.
• Addition and Subtraction - From left to right as well.

In our example, we first focused on solving expressions within brackets, ensuring all steps were tackled in the correct order. Neglecting this order can lead to incorrect results.

Square Roots

Square roots are a way of expressing a number that when multiplied by itself gives the original number. It's noted as √. For example, the square root of 9 is 3, because 3 multiplied by 3 equals 9.
In our original expression, there was a square root - √(25-16). Step by step, we first performed the subtraction within the square root, reducing it to √9, before calculating the root itself.
Understanding square roots is important for tackling problems involving exponents and ensures precision when simplifying complex expressions involving square roots.

Addition and Subtraction

Addition and subtraction are basic math operations but play a key part in simplifying expressions. When dealing with complex brackets or formulas, addition and subtraction require precision to avoid errors.
In our exercise, adding and subtracting inside brackets and across the expression, like \(4+(-9)\) resulting in \(-5\), and later subtracting as in \{29\}-6\, simplified the expression significantly. These operations work together in phases, emphasizing their order and interaction with multiplication and division.

Multiplication and Division

Multiplication (×) and division (÷) are core mathematical operations that probably appear often in various expressions. They are often linked because they carry equal precedence in the order of operations (PEMDAS), which guides us to solve them from left to right.
In the given solution, multiplication was used to manage terms inside braces like transforming \(19-2(-5) = 19 + 10\), while division came into play for simplifying when 18 was divided by 3. Mastering these can result in clarity and accuracy when dealing with complex nested expressions.