Problem 30 - Elementary and Intermediate Algebra

Question

Multiply and simplify. All variables represent positive real numbers. $$ -\sqrt{3}(\sqrt{7}-\sqrt{15}) $$

Step-by-Step Solution

Answer

The simplified expression is \\(-\sqrt{21} + 3\sqrt{5}\\).

Step 1: Distribute the Radical Term

To multiply, distribute the term \(-\sqrt{3}\) through the expression inside the parentheses, which is \(\sqrt{7} - \sqrt{15}\). This results in \(-\sqrt{3} imes \sqrt{7}\) and \(-\sqrt{3} imes (-\sqrt{15})\).

Step 2: Multiply the Radicals

Multiply the radicals by using the property \(\sqrt{a} \times \sqrt{b} = \sqrt{ab}\). For \(-\sqrt{3} \times \sqrt{7}\), calculate \(-\sqrt{21}\). For \(-\sqrt{3} \times (-\sqrt{15})\), which is positive because the negatives cancel, calculate \(+\sqrt{45}\).

Step 3: Simplify the Radicals

Check if the radical expressions can be simplified. \(\sqrt{21}\) cannot be simplified further. \(\sqrt{45}\) can be simplified: as \(45 = 3^2 \times 5\), \(\sqrt{45} = \sqrt{9 \times 5} = 3\sqrt{5}\).

Step 4: Combine the Terms

Combine the terms from Step 2 and Step 3: \(-\sqrt{21} + 3\sqrt{5}\). The expression is now simplified and does not combine further because the radicals are different.

Key Concepts

Simplifying Radicals

Simplifying radicals means reducing a radical expression to its simplest form by removing any possible square factors that are perfect squares. In radical expressions, this means breaking down the number or the variable under the square root sign into its prime factors, allowing us to simplify when possible.
For instance, in the exercise problem, we find the expression \(\sqrt{45}\). This number can be broken down into prime factors: \(45 = 3^2 \times 5\), which allows us to simplify \(\sqrt{45}\) to \(3\sqrt{5}\). This happens because \(\sqrt{9}\) simplifies to 3, which is a perfect square.
This technique not only simplifies making subsequent calculations easier but also frequently allows for situations where radicals can be added or subtracted if like radicals are present. Understanding this concept well aids in solving a wide range of algebraic problems involving roots.

Distributive Property

The distributive property is an essential algebraic rule used when you need to multiply a single term by two or more terms inside a set of parentheses. This property allows you to "distribute" the multiplication over addition or subtraction, ensuring that every term inside the brackets is multiplied by the factor outside. Use this property to expand expressions and simplify equations.
In our original exercise, we deal with the expression \(-\sqrt{3}(\sqrt{7} - \sqrt{15})\). The distributive property tells us that we must multiply \(-\sqrt{3}\) by both \(\sqrt{7}\) and \(-\sqrt{15}\). Each resulting product is calculated separately, yielding \(-\sqrt{3} \times \sqrt{7}\) and \(-\sqrt{3} \times (-\sqrt{15})\), before combining them back together.
This method is invaluable for simplifying complex expressions and equations, allowing algebraic manipulation even with the involvement of radicals.

Multiplying Radicals

Multiplying radicals follows a simple rule: when you multiply two radical expressions, you multiply the numbers or variables inside the radicals, provided they have the same root degree. When multiplying square roots, use the property \(\sqrt{a} \times \sqrt{b} = \sqrt{ab}\).
This step was used to simplify the given expression. Taking \(-\sqrt{3} \times \sqrt{7}\), we calculated \(\sqrt{21}\). Similarly, for \(-\sqrt{3} \times (-\sqrt{15})\), the expression becomes \(\sqrt{45}\).
Keep in mind, the negatives cancel out in the second product, which maintains the product as positive. Multiplying radicals effectively reduces complex expressions, allowing simplified results whenever possible and is an essential skill in algebraic manipulation.