Problem 74 - Contemporary Precalculus

Question

Simplify, and write the given number without using absolute values. $$-|-5|^{2}$$

Step-by-Step Solution

Answer

Answer: The simplified solution of the expression is $$-25$$.

Step 1: Evaluate the Absolute Value

First, let's evaluate the absolute value of -5: $$|-5| = 5$$ So now our expression becomes: $$-|5|^2$$

Step 2: Square the Number

Now let's square the number inside the absolute value brackets: $$5^2 = 25$$ Our expression now looks like this: $$-25$$

Step 3: Final Answer

As there are no additional operations to perform, our final answer is simply: $$-25$$

Key Concepts

Simplifying Expressions

Simplifying expressions involves breaking down an expression into its simplest form. This often means prioritizing certain mathematical operations over others, observing the hierarchy dictated by the order of operations or the BODMAS/BIDMAS rule (Brackets, Orders, Division/Multiplication, Addition/Subtraction).

For example, when simplifying an expression like \(-|5|^{2}\), you perform the following steps:

• First, resolve any operations inside the brackets, which in this case refers to the absolute value.
• Next, deal with the "orders" operation, which includes squaring or other powers.

Additionally, when expressions include absolute values, it is essential to compute them correctly, as they will affect the results of further operations. For a clear understanding, imagine absolute value as a 'distance from zero', which always yields a non-negative outcome. In our example, \(|-5| = 5\). This synthesized approach makes complex problems easier and prepares you to tackle more advanced mathematics.

Squaring Numbers

Squaring a number means multiplying it by itself. This operation is part of "orders" in the order of operations. In the context of simplifying expressions, it is fundamental to accurately evaluate the square before considering any operations outside of it.

For instance, in our example, after evaluating the absolute value we are left with \(5^2\). When we square 5, it becomes \(5 \times 5 = 25\). By executing this step, a 'square' operation turns a positive integer into another positive integer of higher value, depending on the initial digits.

Squaring is always positive because both negative and positive numbers, when multiplied by themselves, result in a positive product. This property is particularly important when working with absolute values and negative numbers, as squaring effectively neutralizes signs.

Negative Sign Operations

Operations involving negative signs can often confuse students, but understanding the basic rules makes simplifying such expressions much easier. In our example expression \(-|5|^{2}\), the negative sign is applied after the evaluation of the absolute value and the square.

To handle the negative sign properly:

• First, ensure that all expressions inside any absolute value or exponent are evaluated.
• Then, apply the negative sign after these evaluations.

In this problem, after simplifying the absolute value and squaring it, we arrived at \(25\). The negative sign preceding the square indicates that the result of the operation is negative, not the square itself. Thus, \(-25\) is the correct simplification.

Remember, a standalone negative sign before a number or expression flips its sign to negative. Understanding these basic principles will help you perform complex calculations confidently and accurately.