Problem 56 - Precalculus Mathematics for Calculus

Question

Perform the indicated operations and simplify. $$(\sqrt{h^{2}+1}+1)(\sqrt{h^{2}+1}-1)$$

Step-by-Step Solution

Answer

The expression simplifies to \(h^2\).

Step 1: Recognize the Expression Form

The expression \((\sqrt{h^{2}+1}+1)(\sqrt{h^{2}+1}-1)\) is in the form of a difference of squares, \((a+b)(a-b) = a^2 - b^2\), where \(a = \sqrt{h^2 + 1}\) and \(b = 1\).

Step 2: Apply the Difference of Squares Formula

Using the difference of squares formula, calculate \((\sqrt{h^2+1})^2 - 1^2\). First, compute \(a^2 = (\sqrt{h^2+1})^2 = h^2 + 1\). Then calculate \(b^2 = 1^2 = 1\).

Step 3: Simplify the Expression

Subtract \(b^2\) from \(a^2\): \(h^2 + 1 - 1\), which simplifies to \(h^2\).

Key Concepts

Simplification of Expressions

Simplification of expressions is a fundamental skill in algebra that involves reducing expressions to their simplest form. The main goal is to make expressions easier to understand or solve. In our exercise, we have the expression \((\sqrt{h^{2}+1}+1)(\sqrt{h^{2}+1}-1)\). To simplify it, we first recognized it as a difference of squares, a common algebraic form. This recognition allows us to apply a straightforward algebraic rule.

When simplifying, it's crucial to follow these steps:

• Identify any patterns or forms, like the difference of squares.
• Use algebraic formulas to make computations easier. These formulas act as shortcuts that prevent the need to multiply each term individually.
• Perform calculations accurately, ensuring each step follows logically from the previous.
• Finally, reduce the expression as much as possible to simplify it further.

In our specific example, we simplified \((h^2 + 1 - 1)\) to \(h^2\) by understanding the structure of the expression and applying the difference of squares formula.

Algebraic Identities

Algebraic identities are pre-established equations that are always true for any variable values. They provide pivotal shortcuts in algebraic manipulations and problem-solving. One of the most significant identities used in our exercise is the difference of squares: \((a+b)(a-b)=a^2-b^2\). Recognizing and applying this identity simplifies what might seem like a complex multiplication problem into a much simpler subtraction problem.

For the expression \((\sqrt{h^{2}+1}+1)(\sqrt{h^{2}+1}-1)\), we applied the identity by setting \(a = \sqrt{h^2 + 1}\) and \(b = 1\). Consequently, the expression transforms into \((\sqrt{h^2+1})^2 - 1^2\), which simplifies to \(h^2 + 1 - 1\). This manipulation demonstrates how powerful algebraic identities simplify complex operations into manageable steps. Using identities like these not only saves time but helps avoid errors during calculations.

Radical Expressions

Radical expressions include square roots and higher-order roots. Simplifying them involves following specific rules that deal with roots and their properties. These expressions can initially look intimidating, but with the right approach, they become manageable.

In our problem, we encountered the radical expression \(\sqrt{h^{2}+1}\). This is transformed through simplification using the difference of squares approach. When squaring the radical, \((\sqrt{h^2+1})^2\), the square root "undoes" itself, simplifying to just \(h^2 + 1\). It’s a helpful property that radicals exhibit: squaring a square root returns the radicand (the part under the root).

Understanding how to manipulate radical expressions is essential in algebra, as it allows us to simplify and solve equations more effectively. Being confident with square roots and their properties will not only make tackling radicals easier but also enhance overall mathematical proficiency.