Problem 43 - Algebra for College Students

Question

Find the following products and express answers in simplest radical form. All variables represent nonnegative real numbers. \((\sqrt{6}+4)(\sqrt{6}-4)\)

Step-by-Step Solution

Answer

The product is -10.

Step 1: Identify the Expression Format

The expression \((\sqrt{6}+4)(\sqrt{6}-4)\) is in the form of \((a+b)(a-b)\), which is a standard difference of squares.

Step 2: Apply the Difference of Squares Formula

The difference of squares formula is \((a+b)(a-b) = a^2 - b^2\). In this problem, identify \(a = \sqrt{6}\) and \(b = 4\).

Step 3: Calculate \(a^2\) (Square the First Term)

Square the first term: \((\sqrt{6})^2 = 6\).

Step 4: Calculate \(b^2\) (Square the Second Term)

Square the second term: \(4^2 = 16\).

Step 5: Subtract \(b^2\) from \(a^2\)

Subtract the square of the second term from the square of the first term: \(a^2 - b^2 = 6 - 16 = -10\).

Step 6: Simplify the Expression

The simplified result of the product is \(-10\).

Key Concepts

Radical Expressions

Radical expressions include roots like square roots, cube roots, and so on. They typically involve numbers or variables under a radical sign (the square root symbol). Understanding radicals is crucial in algebra because they often appear in equations and need to be simplified. In the expression - \(\sqrt{6}+4\)- \(\sqrt{6}-4\)we're using square roots, specifically \(\sqrt{6}\). When working with radicals, it's important to remember that the radicand (the number inside the root) must be nonnegative if we're working within the scope of real numbers. This is why, in exercises like these, nonnegative real numbers are assumed for the variables.

Difference of Squares

The difference of squares is a special algebraic identity that states - \(a^2 - b^2 = (a+b)(a-b)\).This identity can simplify many algebraic expressions or make multiplication quicker when applied.In our example, the expression \(\sqrt{6}+4\) and \(\sqrt{6}-4\) fits perfectly into the - \(a+b\)- \(a-b\)pattern, where \(a = \sqrt{6}\) and \(b = 4\). Knowing that this represents a difference of squares can expedite solving the problem, allowing us to bypass intermediate multiplication steps and jump directly to - \(a^2 - b^2\).

Simplifying Expressions

Simplifying expressions involves reducing them to their most concise form while retaining their value. This often makes expressions easier to work with in calculations or when solving equations.In the given problem, once we recognize that we're working with a difference of squares, we can quickly simplify the expression from - \(\sqrt{6}+4\)- \(\sqrt{6}-4\)by using the formula \(a^2 - b^2\). We compute - \(a^2\) as \(6\)- \(b^2\) as \(16\), giving us- \(6 - 16 = -10\).The final expression is already in its simplest radical form, allowing us to conclude that the simplified result of the original product is indeed - \(-10\).