Problem 22 - College Algebra Essentials

Question

Use the product rule to simplify the expressions in Exercises \(13-22\) In Exercises \(17-22,\) 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: Identify and define the product rule

The product rule of radicals states that the square root of a product equals the product of square roots of individual factors. Therefore, according to this rule, \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\).

Step 2: Apply the product rule

Following the product rule, substitute \(a = 6x\) and \(b = 3x^{2}\) into the equation, \(\sqrt{6x} \cdot \sqrt{3x^{2}} = \sqrt{6x \cdot 3x^{2}}\).

Step 3: Simplify the expression

Multiplication inside the square root gives \(\sqrt{18x^{3}}\) which simplifies to \(3x\sqrt{2x}\).

Key Concepts

Simplifying Radical Expressions

Understanding how to simplify radical expressions is crucial for dealing with more complex algebraic operations. A radical expression is an expression that involves roots, such as square roots, cube roots, etc. To simplify a radical expression, the objective is to make the radical as simple as possible.

The process often involves factoring the number or expression under the radical to reveal perfect squares (or other perfect powers) which can then be taken out of the radical sign. For example, \( \sqrt{18x^3} \) can be broken down because it contains a perfect square, \( 9x^2 \), resulting in \( 3x \sqrt{2x} \).

Another key aspect is to ensure that any variables under the radical are raised to a power that is a multiple of the root (in case of \( \sqrt{} \) the power must be 2, for cube roots, it must be 3, and so on). Simplification may involve rationalizing the denominator, if a radical appears there, to achieve an equivalent expression that has a rational number in the denominator.

Square Root Multiplication

Multiplying square roots follows a specific set of rules stemming from the general principles of exponents and radicals. When two square roots are multiplied together, the product rule of radicals allows us to combine the radicands (the numbers or expressions under the radical signs) before taking the square root.

For instance, given two radical expressions \( \sqrt{a} \) and \( \sqrt{b} \), the product rule of radicals tells us that \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \). Using this rule makes it much easier to multiply radicals and can often lead to further simplification. After applying the rule, as seen in the example \( \sqrt{6x} \cdot \sqrt{3x^2} = \sqrt{18x^3} \) the resulting expression under the radical can often be simplified further, if there are any perfect square factors.

Exponents and Radicals

For a solid understanding of how to manipulate radical expressions, one should be familiar with the connection between exponents and radicals. The square root of a number can be thought of as the number raised to the \( \frac{1}{2} \) power. In general, a number with a fractional exponent \( \frac{m}{n} \) is equivalent to the nth root of the number raised to the mth power.

Properties to Remember

• \( a^{\frac{m}{n}} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m \)
• \( (a^m)^n = a^{m \cdot n} \)
• When simplifying expressions like \( \sqrt{x^2} \), the result is always positive (or 0), since x is assumed to be a nonnegative real number.

Understanding exponents is essential when simplifying radical expressions, as you often need to express radicals as powers to simplify or combine like terms.

Nonnegative Real Numbers

In mathematics, especially when dealing with radicals and exponents, we often restrict our focus to nonnegative real numbers. These include all positive real numbers and zero. The restriction is significant since the even root of a negative number is not real.

When variables are assumed to represent nonnegative real numbers, we avoid dealing with the complexities of complex numbers (which involve the square root of a negative number). This simplification is important for the assumptions made when simplifying expressions involving variables inside of radicals, as demonstrated in problems like \( \sqrt{6x} \cdot \sqrt{3x^2} \), where x is assumed to be a nonnegative real number to ensure the result is also a real number.