Problem 16 - Algebra 2

Question

Simplify. $$ \sqrt{(-18)^{2}} $$

Step-by-Step Solution

Answer

The simplified form is 18.

Step 1: Recognize the Mathematical Structure

The expression given is \( \sqrt{(-18)^{2}} \), which can be translated to the square root of a squared number. The expression inside the square root, \((-18)^{2}\), is a perfect square since it's a number raised to the power of 2.

Step 2: Simplify the Squared Term

Calculate the square of \(-18\). When you square a negative number, the result is always positive. Therefore, \((-18)^{2} = 324\). This simplifies the expression to \( \sqrt{324} \).

Step 3: Evaluate the Square Root

Now, find the square root of 324. The square root of 324 is 18 since \(18 \times 18 = 324\). So, \( \sqrt{324} = 18 \).

Step 4: Address the Absolute Value Aspect

Even though \( -18 \) was the original number, once squared, the sign is lost, and thus we consider the principal root (which is always non-negative) when simplifying a square root. Therefore, the final answer is \(|-18| = 18\).

Key Concepts

Square Roots

Square roots are operations that help us find a number which, when multiplied by itself, gives the original number. For instance, the square root of 16 is 4, because when 4 is multiplied by 4, it results in 16. The symbol for a square root is \( \sqrt{} \). Square roots can pertain to both positive and negative numbers, but in mathematics, the principal square root is always the non-negative root. In our exercise, we dealt with \( \sqrt{324} \), which equals 18, because \( 18 \times 18 = 324 \). It's crucial to identify square roots especially when simplifying expressions to their simplest form.

Perfect Squares

A perfect square is a number that is the result of squaring a whole number. For example, 1, 4, 9, and 16 are all perfect squares because they can be achieved by squaring 1, 2, 3, and 4, respectively. In the given exercise, \((-18)^{2}\) is a perfect square because it simplifies to 324, which is the square of 18. Perfect squares are significant because they produce integer square roots, making them simpler to work with. Recognizing perfect squares in an expression can greatly aid in simplifying the expression efficiently.

Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. This means it is always a non-negative number. The absolute value is denoted by vertical bars, like \( |x| \). For example, the absolute value of both 5 and -5 is 5. In the exercise, once we calculated \( (-18)^{2} = 324 \), taking the square root effectively left us with the absolute value of -18. Hence, \( |-18| = 18 \). This is because considering the principal root means we take the non-negative value.

Negative Numbers

Negative numbers are less than zero and are often represented with a minus sign. They play a key role in various mathematical operations. When you square a negative number, such as -18, the negative becomes positive because multiplying two negative numbers results in a positive product. For example, \( (-18) \times (-18) = 324 \). This transformation is crucial while simplifying expressions that involve squaring negative numbers. A consistent understanding of when and how negative numbers affect outcomes is essential for correctly solving equations and simplifying expressions.