Problem 20 - College Algebra with Modeling and Visualization

Question

Evaluate by hand. $$ (4-5)^{2}-3^{2}-3 \sqrt{9} $$

Step-by-Step Solution

Answer

The value of the expression is -17.

Step 1: Simplify the Expression Inside the Parentheses

First, evaluate the expression inside the parentheses. Calculate:\(4 - 5 = -1\)

Step 2: Square the Result of the Parentheses

Take the result from Step 1, which is \(-1\), and square it:\((-1)^2 = 1\)

Step 3: Evaluate the Square Root

Calculate the square root in the expression \(3 \sqrt{9}\):Since \(\sqrt{9} = 3\), we have:\(3 \times 3 = 9\)

Step 4: Calculate the Square of a Number

Find \(3^2\) which is part of the expression:\(3^2 = 9\)

Step 5: Substitute Back into the Original Expression

Now you will substitute back all the calculated values into the original expression:The expression becomes:\(1 - 9 - 9\)

Step 6: Simplify the Final Expression

Perform the arithmetic operations from left to right:\(1 - 9 = -8\)Then:\(-8 - 9 = -17\)

Key Concepts

Evaluating Expressions

Evaluating expressions is the process of finding the value of a mathematical statement. It involves simplifying each part of the equation or expression, often using calculations of constants, variables, or operations. In this exercise, we start by handling specific components within the expression: parentheses, exponents, and square roots. Begin by evaluating the expression inside the parentheses, as dictated by the order of operations, to transform the initial part of the equation into a simpler form. This part of the process can remove complexity from the expression and make it easier to calculate subsequent steps.

Order of Operations

The order of operations is crucial for evaluating expressions correctly. Known by the acronym PEMDAS (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction), it dictates the sequence in which operations should be performed to ensure accurate results. In our example:

• Parentheses are evaluated first, so we calculate \(4 - 5\) to get \(-1\).
• Then, handle exponents: square \(-1\) to get \(1\), and separately find \(3^2 = 9\).
• Next, any remaining operations are carried out: in this case, square roots and multiplication within \(3\sqrt{9}\).
• Finally, subtraction and further calculation simplify the expression to its final result \(1 - 9 - 9 = -17\).

By consistently applying these rules, one ensures that the expression is simplified correctly, without any mathematical errors due to incorrect order.

Square Roots

Square roots are a type of radical expression which inverse the operation of squaring a number. Evaluating square roots involves finding a number that, when multiplied by itself, results in the original number. For instance, the square root of 9 is 3 because \(3 \times 3 = 9\). In the given expression, recognizing that \(\sqrt{9} = 3\) allows us to further simplify the expression by multiplying \(3 \times 3\) to get 9. Understanding square roots is essential as they frequently appear in algebraic expressions and demand careful attention to simplify properly without ambiguity. This skill of evaluating square roots is fundamental in many mathematical problems, especially in algebra and geometry.