Problem 44 - Calculus with Concepts in Calculus

Question

Evaluate the expression. $$ |-\sqrt{2}|^{2} $$

Step-by-Step Solution

Answer

2

Step 1: Simplify Inner Expression

First, write down the inner expression inside the absolute value. Here, it is -\(-\sqrt{2}\). Since the expression outside the square root is negative, apply the negative sign to the square root term to get \(-\sqrt{2}\).

Step 2: Evaluate the Absolute Value

Take the absolute value of \(-\sqrt{2}\). The absolute value of a negative number makes it positive. Therefore, \(|-\sqrt{2}| = \sqrt{2}\).

Step 3: Square the Result

Now, take the expression obtained from the absolute value and raise it to the power of 2. \(\left(\sqrt{2}\right)^{2}\) equals 2, since squaring a square root returns the number inside the square root.

Key Concepts

Properties of Square Roots

Understanding the properties of square roots is fundamental for solving problems involving them. A square root, denoted as \( \sqrt{x} \), is a number that, when multiplied by itself, gives the original number \( x \). This property is useful when simplifying expressions or solving equations. For instance:

• The square root of a number is always non-negative. If \( x \) is non-negative, \( \sqrt{x} \) is also non-negative.
• The expression \( \sqrt{x^2} \) simplifies to \( |x| \). This is because the square of a negative or positive number is always positive, and taking the square root of that squared result will yield the absolute value of the original number.

These properties help us manipulate and understand more complex expressions by letting us simplify them in consistent ways, ensuring that expressions are evaluated correctly.

Negative Numbers in Expressions

When dealing with negative numbers, especially within expressions that involve absolute values and other operations, it is vital to remember a few key points:

• The absolute value of a number, denoted as \( |x| \), is the distance of that number from zero. This means it is always non-negative, no matter the original sign of \( x \). Taking the absolute value of a negative number makes it positive.
• Operations involving negatives, such as subtraction and division, follow specific rules. For example, subtracting a negative is equivalent to adding a positive.
• Expressions can often involve multiple operations with negative numbers, such as multiplication. Here, two negatives make a positive, while a negative and a positive make a negative.

These concepts are important for navigating through expressions that might look complicated at first glance. They allow for simplifying steps without changing the meaning or value of the expressions encountered.

Powers and Exponents

Powers and exponents are tools that represent repeated multiplication. Understanding how they work is crucial for simplifying expressions and solving algebraic problems. Let's look at some basic principles:

• An exponent like \( a^n \) means that the number \( a \) is multiplied by itself \( n \) times. For example, \( 2^4 = 2 \times 2 \times 2 \times 2 = 16 \).
• When a number is raised to the power of 2, it is termed as "squared". For example, \( x^2 \) indicates \( x \) multiplied by itself: \( x \times x \).
• The inverse operation of squaring is the square root. For instance, if \( (\sqrt{x})^2 = x \), then squaring the square root undoes the square root operation.

Recognizing these patterns and applying properties of powers and exponents allows for breaking down complex expressions into more manageable parts, especially when combined with other mathematical operations.