Problem 65 - Algebra 1: Concepts and Skills

Question

Evaluate the radical expression when $a=-1 \text { and } b=5. $$\sqrt{a^{2}-1}$$

Step-by-Step Solution

Answer

The evaluated value of the expression \( \sqrt{a^2 - 1} \) when a = -1 is 0.

Step 1: Substitute the value of 'a' into the equation

Firstly, replace 'a' with its given value which is -1. So the expression becomes \( \sqrt{(-1)^2 - 1} \)

Step 2: Evaluate the power first

According to the order of operations (PEMDAS/BODMAS), powers are done before subtraction. Thus, evaluate the power (-1)^2. The expression after evaluating the power becomes \( \sqrt{1 - 1} \)

Step 3: Perform subtraction

Next, perform the subtraction operation as per order of operations (PEMDAS/BODMAS). When we subtract 1 from 1, we get 0. Thus, the expression becomes \( \sqrt{0} \)

Step 4: Evaluate the square root

Finally, evaluate the square root. The square root of any number 'n' is the value that, when multiplied by itself, gives 'n'. The square root of 0 is simply 0. Therefore, final answer is 0.

Key Concepts

Understanding the Order of Operations

When solving mathematical expressions, following the correct order is crucial. This is where the Order of Operations, often remembered by the acronym PEMDAS or BODMAS, comes into play. PEMDAS stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). BODMAS is similarly used, where B represents Brackets and O symbolizes Order (or exponents like squares and square roots).

In our exercise, this rule helps us determine the sequence to evaluate each part of the expression \( \sqrt{(-1)^2 - 1} \).

• Powers first: Evaluate \((-1)^2\), which is 1.
• Subtraction second: Then perform \(1 - 1\) to get 0.
• Finally, evaluate the square root.

Skipping or rearranging this order can lead to incorrect results.

Simplifying Through Substitution

Substitution is a valuable technique in mathematics, which allows for simplifying complex expressions by replacing variables with given values.

In the original exercise, we are given the value \(a = -1\). By substituting this into the expression \(\sqrt{a^{2} - 1}\), it becomes clearer and more manageable.

• Replace \(a\) with \(-1\) to transform the expression into \(\sqrt{(-1)^2 - 1}\).
• This substitution helps us focus on evaluating the operations without worrying about variable manipulation.

Substitution is especially useful for breaking down problems and ensuring each part of the expression is accurate.

Evaluating the Square Root

Square root evaluation involves finding a number that, when multiplied by itself, gives the original number. It's one of the fundamental operations in mathematics, often represented by the radical symbol \( \sqrt{} \).

In our scenario, we reach the expression \( \sqrt{0} \) after performing the previous operations.

• The square root of 0 is 0 because 0 multiplied by itself is 0.
• Understanding square roots is essential, especially in many fields like geometry and algebra.

Practicing square root operations helps build a foundation for more complex mathematical concepts, including quadratic equations and geometry.