Problem 74 - Intermediate Algebra

Question

Evaluate each expression. See Example \(9 .\) $$ (-1-\sqrt{144})^{2} $$

Step-by-Step Solution

Answer

The expression evaluates to 169.

Step 1: Simplify the Square Root

First, identify and simplify the square root expression inside the parentheses. We have \(\sqrt{144}\). Since 144 is a perfect square, \(\sqrt{144} = 12\). Therefore, rewrite the expression as \((-1 - 12)^2\).

Step 2: Evaluate the Expression Inside the Parentheses

Now, simplify the expression inside the parentheses. We have \(-1 - 12\), which equals \(-13\). So, the expression becomes \((-13)^2\).

Step 3: Square the Result

Finally, evaluate \((-13)^2\). Squaring a negative number: \((-13) \times (-13) = 169\). Thus, the expression \((-1 - \sqrt{144})^2\) is 169.

Key Concepts

Square Root

The square root is a value that, when multiplied by itself, gives the original number. Imagine we have the number 144. We want to find a number that when multiplied by itself equals 144. That number is called the square root of 144.
A square root is represented using the symbol \( \sqrt{} \). For example, \( \sqrt{144} \) is 12, because 12 multiplied by 12 gives 144.

• To find a square root, you can look for a number that gives the original number as a product of itself.
• Square roots are often used to simplify expressions and solve equations.

When dealing with square roots, it's important to determine if the number is a perfect square, which makes the solution straightforward.

Perfect Square

A perfect square is a number that is the square of an integer. In simpler terms, it is a number that you get by multiplying an integer by itself. This means it has an exact whole number as its square root.
For example, 144 is a perfect square because it comes from multiplying 12 by itself: \(12 \times 12 = 144\). So, 12 is a whole number and \( \sqrt{144} = 12 \).

• Knowing what makes a number a perfect square can help in simplifying mathematical problems.
• When you see a number like 144, it's worth checking if any whole number multiplied by itself gets you that number.

Recognizing perfect squares makes calculating square roots much easier, especially when solving problems that involve squared expressions.

Negative Numbers

Negative numbers are numbers less than zero, indicated by a minus (−) sign. They are the opposite of positive numbers and are used to represent various concepts like debt, temperatures below zero, or elevations below sea level.
In mathematics, negative numbers follow certain rules when they are added, subtracted, multiplied, or divided. For example, when you subtract 12 from -1, you move further into the negatives: \(-1 - 12 = -13\).
When squaring negative numbers, remember:

• The product of two negative numbers is positive: \((-13) \times (-13) = 169\).
• Squaring a number means multiplying the number by itself.

This is key when working with squared expressions—as in our example—because the square of a negative number always ends up as a positive result.