Problem 16 - 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. $$\sqrt{125 x^{2}}$$

Step-by-Step Solution

Answer

The simplified form of the expression \(\sqrt{125 x^{2}}\) is \(5x\sqrt{5}\).

Step 1: Identification

Identify the terms inside the square root, \(125\) and \(x^{2}\). Both are perfect squares.

Step 2: Breaking down the square roots

Separate the square root of each term. This stems from the rule \(\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}\). Hence, we have \(\sqrt{125} \cdot \sqrt{x^{2}}\).

Step 3: Simplification

Calculating the square root of each term separately, we have: \( \sqrt{125} = 5\sqrt{5} \) and the \(\sqrt{x^{2}} = x\) (because we are assuming x to be a nonnegative real number according to the exercise instructions).

Key Concepts

Square Roots

Square roots are a mathematical operation that seeks to find a number which, when multiplied by itself, will equal the original number. The symbol used to represent a square root is \( \sqrt{\cdot} \). For instance, the square root of 9 is 3, because \( 3 \times 3 = 9 \). The concept of square roots is especially useful in simplifying expressions, as it allows us to break down more complex numbers into simpler factors.

When dealing with square roots, especially in algebraic expressions, it is often useful to express the square root of a product of variables and numbers separately, which is derived from the product rule for square roots. The product rule states:

• \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \)

This rule allows us to find the square root of each factor individually. This can make the process of simplification easier and more manageable.

Simplification

Simplification is the process of reducing a mathematical expression to its simplest form, making it easier to work with. In algebra, simplification often involves factoring, canceling terms, and applying rules like the product rule for square roots.

For the expression \( \sqrt{125 x^2} \), simplification involves:

• Breaking down \( \sqrt{125} \) into \( 5\sqrt{5} \). This is because 125 can be factored into \( 5 \times 5 \times 5 \), which simplifies to \( 5 \times \sqrt{5} \).
• Recognizing that \( \sqrt{x^2} = x \), given that \( x \) is a nonnegative real number.

After applying these steps, the expression \( \sqrt{125 x^2} \) simplifies to \( 5x\sqrt{5} \). Simplifying expressions not only helps in performing operations but also aids in understanding and solving further algebraic problems.

Nonnegative Real Numbers

Nonnegative real numbers include all positive numbers and zero, essentially every number that is not negative. These numbers are frequently assumed in algebraic expressions to avoid complexities related to negatives and arithmetic involving undefined operations.

In the context of square roots, assuming variables are nonnegative helps us simplify expressions such as \( \sqrt{x^2} \) easily. This is because for nonnegative \( x \), the expression \( \sqrt{x^2} \) simplifies directly to \( x \) without needing to consider additional cases or absolute value symbols.

• Nonnegative numbers range from 0 to positive infinity: \( [0, \infty) \).

Understanding that the variable represents a nonnegative real number streamlines the simplification process, ensuring clarity and correctness in mathematical solutions. By knowing these constraints, students can approach problems with confidence, focusing on applying straightforward algebraic principles.