Question

For Exercises \(1-74,\) simplify. $$\\{12[-2-(-5)]-40\\} \div \sqrt{25-9}$$

Short answer

Answer: -1

Step-by-step solution

Simplify the inner expression [-2-(-5)]

Add 5 to -2: \([-2+5]=3\).

Multiply the result by 12 and subtract 40

Multiply 3 by 12: \(12(3)=36\). Then subtract 40: \(36-40=-4\).

Calculate the square root

Find the difference between 25 and 9: \(25-9=16\). Then calculate the square root of 16: \(\sqrt{16}=4\).

Divide the result by the square root

Divide -4 by 4: \( \frac{-4}{4} = -1\). The simplified expression is -1.

Key concepts

Understanding Order of Operations

The order of operations is a mathematical rule used to determine the sequence in which different operations (such as addition, subtraction, multiplication, and division) should be performed. This is crucial when solving complex expressions to ensure the correct result. Suppose we have a mathematical expression with different operations; how do we know which part to solve first?

• Parentheses or brackets are always solved first. They signify that the operations contained within should be treated as one unit.
• Exponentiation and square roots are tackled next.
• Multiplication and division (from left to right) follow.
• Addition and subtraction (from left to right) are handled last.

A useful way to remember this order is the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division, Addition and Subtraction. Following this order ensures that expressions are simplified correctly and efficiently.
In the original exercise, you first worked with the expression inside the brackets: \[-2-(-5) \rightarrow [-2 + 5] = 3\] This was then followed by multiplying, subtracting, computing the square root, and finally division.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form. This ensures that mathematical expressions are easier to read and understand. It is especially important when dealing with more complicated expressions, as it helps to see the underlying structure clearly. Let's see the examples of simplification from the exercise:

• First, solve all expressions within any parentheses or brackets to eliminate them.
• Combine like terms and perform all necessary arithmetic operations.
• Simplify square roots and exponents.
  For instance, in the original exercise, after simplifying \[-2 - (-5)\] to \[3\], multiplication and subtraction followed: \[12(3) - 40\] gives \[-4\].

By breaking down the expression step-by-step, the process of simplification becomes straightforward. Always evaluate each part of the expression in sequence to arrive at the simplest form.

Delving into Square Roots

Square roots are fundamental in mathematics, representing a value that, when multiplied by itself, yields the original number. The symbol for square root is \(\sqrt{}\). Calculating square roots is an essential skill when simplifying expressions, as shown in the original exercise.
Consider the square root operation in the provided example: \[ \sqrt{16} = 4. \]You first had to find the difference: \[25 - 9 = 16\], then calculate the square root of that result. Finding square roots is simple when the number is a perfect square, like 16, 25, or 9.

• If the number under the square root is not a perfect square, an approximate value can be used or expressed in simplest radical form.
• Square roots are used in various computations, including in the Pythagorean theorem and quadratic equations.

Understanding how to handle square roots helps in comprehending more complex algebraic operations.