Problem 18 - 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. $$\sqrt{10 x} \cdot \sqrt{8 x}$$

Step-by-Step Solution

Answer

The simplified expression is \( 4x\sqrt{5} \).

Step 1: Analyze and use the product rule of square roots

The product rule of square roots can be applied to the given expression \( \sqrt{10x} * \sqrt{8x} \). This rule states that \( \sqrt{a} * \sqrt{b} = \sqrt{a*b} \). Therefore, apply this rule to the given expression: \( \sqrt{10x} * \sqrt{8x} = \sqrt{(10x) * (8x)} \).

Step 2: Compute the product

Calculate the product inside the square root: \( 10x * 8x = 80x^2 \). So the new expression is \( \sqrt{80x^2} \).

Step 3: Simplify the square root

Break down the number inside the square root into its prime factors and find the square root of \( 80x^2 \). The number \( 80 \) breaks down into \( 2^4*5 \) and \( x^2 \) is simply \( x^2 \). The square root of \( 2^4*5x^2 \) yields \( 2^2*x* \sqrt{5} = 4x\sqrt{5} \).

Key Concepts

Simplifying Expressions

Simplifying expressions can seem complex at first, but it's really about making numbers and variables easier to work with. When you simplify an expression, you aim to present it in its simplest form.

For instance, if you have a product of square roots like \( \sqrt{10x} \cdot \sqrt{8x} \), you can use rules of algebra and arithmetic to simplify it into a form that's easier to understand or solve. Simplification often involves factoring numbers, combining like terms, or using mathematical rules—such as the product rule of square roots.

This helps you focus on the core components without unnecessary complexity. Feeling comfortable with simplifying can make algebraic and numerical problems less daunting.

Nonnegative Real Numbers

Real numbers encompass all the numbers on the number line, both positive and negative, as well as zero. However, nonnegative real numbers are a subset that includes all positive real numbers and zero.

When dealing with square roots, particularly in algebraic expressions, assuming or specifying that variables represent nonnegative real numbers is commonly the case. This is because the square root function is usually defined for nonnegative numbers in real number arithmetic.

When solving problems like \( \sqrt{10x} \cdot \sqrt{8x} \), it ensures the expressions are not undefined or imaginary, simplifying the process of interpretation and computation. Nonnegative assumptions make it permissible to effortlessly apply rules, such as moving square roots around multipliers.

Square Roots

The square root of a number is a value that, when multiplied by itself, gives the original number. Square roots are denoted by the radical symbol \( \sqrt{} \), and for any nonnegative real number \( a \), \( \sqrt{a} \) is also a nonnegative real number.

When working with products of square roots such as \( \sqrt{10x} \cdot \sqrt{8x} \), the product rule is a powerful tool. This rule \( \sqrt{a} \cdot \sqrt{b} = \sqrt{ab} \) allows you to combine roots into a single square root for simplification.

Understanding the properties of square roots and their rules can significantly simplify calculations in algebraic expressions, making complex expressions concise and manageable.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They're fundamental in expressing mathematical ideas and relationships.

For example: \( \sqrt{10x} \cdot \sqrt{8x} \). This expression combines constants (like 10 and 8) with a variable (\( x \)). By applying algebraic principles such as the product rule of square roots, you can reorganize or simplify these expressions.

Through simplification, the calculation becomes \( 4x\sqrt{5} \), making it easier to further analyze or use in equations. Mastery of expressions involves learning to manipulate these components fluidly, aiding in problem solving across mathematics.