Problem 37 - Algebra 2 and Trigonometry

Question

In \(3-41\) , express each product in simplest form. Variables in the radicand with an even index are non-negative. $$ (a+\sqrt{b})(a-\sqrt{b}) $$

Step-by-Step Solution

Answer

The simplest form is \(a^2 - b\).

Step 1: Recognize the Expression Type

The expression \((a+\sqrt{b})(a-\sqrt{b})\) is in the form of a difference of squares. This is because we have two binomials, each containing identical terms but with opposite signs between the terms.

Step 2: Apply the Difference of Squares Formula

The formula for the difference of squares is \((x+y)(x-y) = x^2 - y^2\). Here, \(x = a\) and \(y = \sqrt{b}\). Substituting into the formula gives us:\[(a+\sqrt{b})(a-\sqrt{b}) = a^2 - (\sqrt{b})^2\]

Step 3: Simplify the Squared Term

Since \((\sqrt{b})^2 = b\), we can simplify the expression further:\[a^2 - b\]

Step 4: Confirm the Simplified Form

The expression \(a^2 - b\) is the simplest form of the given product because there are no like terms to combine or further factors to simplify. Thus, the expression is fully simplified.

Key Concepts

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra. It involves reducing an expression to its simplest form while maintaining its value. When simplifying expressions, we aim to make them easier to understand and solve.

In this case, we began with the expression \((a+\sqrt{b})(a-\sqrt{b})\). By identifying it as a difference of squares, which is a special product, we used a specific formula to simplify the process. This formula is \((x+y)(x-y) = x^2 - y^2\).

By applying this formula, we repackaged the multiplication of two binomials into a subtraction of squares, namely \(a^2 - b\). This continues the focus on simplifying into the cleanest, most digestible form possible. Simplification often leads to quicker solutions and deeper understanding of the mathematical relationships between terms.

Radicands

A radicand is the number or expression located under the radical sign (square root, cube root, etc.). In our expression, \(\sqrt{b}\), the letter \(b\) is the radicand. It is essential to understand how radicands interact with each other, especially under a radical, to fully simplify expressions that involve them.

The nature of the radicand can affect how we simplify the entire expression. In situations where the radicand holds variables, we must consider the rules that apply to even and odd indices—like whether variables are non-negative.

In our example, the task specified that variables under even indices are non-negative, which ensures that \((\sqrt{b})^2 = b\) without complications that might arise from potential negative values. Recognizing these conditions simplifies dealing with the radicals in expressions.

Binomials

A binomial is an algebraic expression containing exactly two terms. In the given problem, both \((a+\sqrt{b})\) and \((a-\sqrt{b})\) are binomials. These expressions feature an addition or subtraction operation between their terms.

The arrangement of the terms with opposing signs—one positive and one negative—is particularly noteworthy here. This configuration is what allows us to use the difference of squares technique to simplify the expressions.

The careful observation of binomials with oppositely-signed terms is crucial in recognizing opportunities to apply specific formulas, like the difference of squares. It highlights how understanding the structural properties of an expression can impact the methods used in solving it.