Problem 16 - College Algebra

Question

Use the product rule to simplify the expressions in Exercises \(7-16 .\) In Exercises \(11-16,\) assume that variables represent nonnegative real numbers. $$\sqrt{6 x} \cdot \sqrt{3 x^{2}}$$

Step-by-Step Solution

Answer

The simplified form of the expression \(\sqrt{6x} \cdot \sqrt{3x^{2}}\) is \(3x\sqrt{2x}\).

Step 1: Apply the Product Rule

First, apply the product rule of radicals by multiplying the quantities inside the square roots: \(\sqrt{6x} \cdot \sqrt{3x^{2}} = \sqrt{(6x)(3x^{2})}\)

Step 2: Simplify the Expression

Next, simplify the expression inside the square root. Multiply the numbers and the variables separately: \(\sqrt{(6x)(3x^{2})} = \sqrt{18x^{3}}\)

Step 3: Break Down the Square Root

The expression can be further simplified by breaking down the square root into simpler radicals. \(x^{3}\) can be rewritten as \(x \cdot x^{2}\) and the square root of 18 can be simplified to \(3\sqrt{2}\). So, \(\sqrt{18x^{3}}\) can be written as \(\sqrt{(9\cdot2) (x^{2}\cdot x)}\) or \(3x\sqrt{2x}\).

Key Concepts

Simplifying Square Roots

Understanding how to simplify square roots is essential for solving problems involving radical expressions. A square root is a number that, when multiplied by itself, gives the original number. Simplifying a square root involves breaking it down into two types of factors - perfect squares and those that are not. Perfect squares are numbers like 1, 4, 9, 16 and so on, which have an integer as their square root.

When simplifying \( \sqrt{18x^3} \), we look for the largest perfect square factor of 18, which is 9. This gives us 9 (a perfect square) times 2 (not a perfect square). For \(x^3\), we realize \(x^2\) is the perfect square factor, leaving us an extra \(x\) that's not a perfect square. This process allows us to rewrite the original square root as a product of square roots, \(3x\sqrt{2x}\), making the numbers inside the radical as small as possible.

Understanding Radical Expressions

Radical expressions, such as \(\sqrt{6x}\cdot\sqrt{3x^2}\), contain a root symbol and represent the root of a quantity. The challenge with these expressions is that they often need to be simplified to perform addition, subtraction, multiplication, or division.

Simplifying a radical expression helps us combine like terms and makes mathematical operations more manageable. In our example, the expression \(\sqrt{6x}\cdot\sqrt{3x^2}\) is a product of radicals. We can combine these under a single radical as \(\sqrt{(6x)(3x^2)}\), setting up the expression to be simplified further as seen in the steps of the solution.

Multiplication of Radicals

Multiplying radicals can seem complicated, but it's like multiplying any other algebraic expression, especially when using the product rule for radicals. The product rule states that the product of two square roots is the square root of the product of the two radicands (the numbers under the radical sign).

For example, when multiplying \(\sqrt{6x}\) and \(\sqrt{3x^2}\), apply the product rule to get \(\sqrt{6x \cdot 3x^2}\). Oh, and a friendly tip: always look to simplify before and after the multiplication. This makes dealing with radical expressions easier and helps with checking your work for any possible mistakes.