Problem 22 - College Algebra

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 final simplified expression is \(3 \sqrt{2}x^{2} \sqrt{x}\).

Step 1: Apply the product rule

According to the product rule for square roots, \(\sqrt{a} \cdot \sqrt{b}\) = \(\sqrt{a \cdot b}\). Apply this rule to the given expression to get \(\sqrt{6x} \cdot \sqrt{3x^{2}} = \sqrt{(6x) \cdot (3x^{2})}\).

Step 2: Simplify the expression inside the square root

Multiply the numbers and the variables separately to simplify the expression. Giving: \( \sqrt{(6 \cdot 3) \cdot (x \cdot x^{2})} = \sqrt{18x^{3}} \).

Step 3: Final simplification

The square root of 18x can be simplified. Write 18 as 2*9, and x^3 as x*x^2. Then take the square root separately giving: \( \sqrt{2 \cdot 9 \cdot x \cdot x^{2}} = \sqrt{2} \cdot 3 \cdot x \cdot \sqrt{x^{2}} = 3 \sqrt{2}x^{2} \sqrt{x} \).

Key Concepts

Simplifying Expressions

Simplifying expressions is a key skill in algebra that helps make calculations easier and more manageable. The goal is to reduce an expression to its simplest or most efficient form. The original expression provided: \(\sqrt{6x} \cdot \sqrt{3x^{2}}\) can initially look complex, but by applying rules such as the product rule for square roots, this task becomes much simpler.

Steps involved in simplifying include:

• Identify parts of the expression that can be combined or reduced.
• Apply algebraic rules correctly, such as the product rule for square roots.
• Perform operations carefully, like multiplication of coefficients and adding of exponents.

The ability to simplify expressions allows us to better understand and manipulate algebraic equations, which is crucial for solving real-world problems and understanding mathematical concepts in depth.

Square Roots

Square roots are mathematical symbols that denote the principal square root of a number or expression. When you see the square root, \(\sqrt{}\), it represents a value that, when multiplied by itself, gives the original number or expression inside the root.

In our example, we're combining two square root expressions: \(\sqrt{6x}\) and \(\sqrt{3x^{2}}\). By utilizing the product rule, which states that \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\), these can be combined into a single square root: \(\sqrt{18x^{3}}\).

Remember:

• Square root simplification often involves identifying perfect squares within the radicand (the number inside the square root).
• Consider breaking down numbers to their prime factors to simplify more effectively.

Square roots are fundamental in mathematics, supporting various calculations in geometry, algebra, and beyond.

Algebraic Expressions

Algebraic expressions involve numbers, variables, and operations like addition, multiplication, and exponentiation. In the expression \(\sqrt{6x} \cdot \sqrt{3x^{2}}\), both numerical coefficients and variable parts are present. Understanding how to manipulate these expressions is invaluable.

Key considerations for dealing with algebraic expressions such as this involve:

• Multiplying coefficients as regular numbers.
• Applying the rules of exponents, such as \(x \cdot x^{2} = x^{1+2} = x^{3}\).
• Simplifying after applying specific rules like the product rule for square roots.

Simplifying algebraic expressions is essential for solving equations, factoring, and understanding the relationships between variables and constants in many mathematical contexts.

Nonnegative Real Numbers

Nonnegative real numbers are numbers that are either positive or zero. In mathematics, they are crucial because certain expressions, particularly those involving square roots, only yield real (non-imaginary) results when all variables are nonnegative.

Why nonnegative numbers matter:

• They ensure that operations like taking square roots remain within the realm of real numbers.
• They help simplify problem-solving by avoiding complex solutions that arise with negative numbers under square roots.
• In contexts like geometry and physics, nonnegative numbers often denote realistic measurements such as length, area, or volume.

Understanding the concept of nonnegative real numbers ensures that you correctly interpret and solve many mathematical problems, while also making sure the solutions are applicable in practical situations.