Problem 28 - 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. $$ \sqrt{5 a}(\sqrt{5 a}-3) $$

Step-by-Step Solution

Answer

The simplest form is \(5a - 3\sqrt{5a}\).

Step 1: Distribute the First Term

We start by distributing \( \sqrt{5a} \) to each term inside the parenthesis \((\sqrt{5a} - 3)\). This step uses the distributive property: \( a(b+c) = ab + ac \).

Step 2: Multiply the First Pair

Calculate \( \sqrt{5a} \times \sqrt{5a} \). The square root of a value times itself equals the original value, so this becomes \( 5a \). Thus, we have \( 5a \).

Step 3: Multiply the Second Pair

Now calculate \( \sqrt{5a} \times -3 \). Multiply the terms together to get \( -3\sqrt{5a} \).

Step 4: Write the Expression in Simplest Form

Combine the results: \( 5a - 3\sqrt{5a} \). This is the simplest form of the expression because no like terms can be combined.

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that helps us expand expressions. It's like a rule that lets you "distribute" one term across terms inside parentheses. The basic form of the distributive property is expressed as \( a(b+c) = ab + ac \). This means you multiply the term on the outside, \(a\), by each term inside the parentheses separately.

In this exercise, we use the distributive property to expand \( \sqrt{5a}(\sqrt{5a} - 3) \). It requires you to:

• Multiply \( \sqrt{5a} \) by \( \sqrt{5a} \).
• Then, multiply \( \sqrt{5a} \) by \(-3\).

By applying the distributive property, we get each pair of terms multiplied, making it easier to simplify the expression later. When you break down a problem using this method, it becomes much more straightforward, allowing for an easier path to finding the simplest form of the expression.

Square Roots

Square roots are one of the first types of roots you will learn in mathematics. Essentially, the square root of a number is a value that, when multiplied by itself, gives the original number. For instance, the square root of 9 is 3, because \(3 \times 3 = 9\).

When dealing with square roots of variables, like \( \sqrt{5a} \), it follows the same logic. Here, \( \sqrt{5a} \times \sqrt{5a} = 5a \), because the square root cancels with itself resulting in the original radicand (the value inside the square root).

Remember these key points about square roots:

• The square root of a product (like \(5a\)) can be split across the factors: \( \sqrt{5a} = \sqrt{5} \times \sqrt{a} \).
• Square rooting reverses the operation of squaring a number, so \( (\sqrt{x})^2 = x \).

This insight is crucial, especially when simplifying algebraic expressions involving radicals, like our current exercise.

Multiplication of Radicals

Multiplying radicals might seem complex at first, but with practice, it becomes second nature. The rule for multiplying radicals involves applying the product rule of square roots: \( \sqrt{a} \times \sqrt{b} = \sqrt{ab} \). This tells us that we can multiply the values inside the radicals and then take the square root of the resulting product.

In our given problem, the two products calculated using radicals are:

• \( \sqrt{5a} \times \sqrt{5a} = \sqrt{(5a) \times (5a)} = \sqrt{25a^2} = 5a \).
• \( \sqrt{5a} \times -3 = -3\sqrt{5a} \).

To approach these problems, apply the multiplication rule to handle each pair of radicals, keeping the coefficients intact and simplifying where possible. This insight helps provide a clear pathway from a problem set with composite radicals to a simplified expression, which is exactly what we did in the exercise: reaching the simplest form \( 5a - 3\sqrt{5a} \).