Use the product rule to simplify the expressions in Exercises 13–22. In Exercises 17–22, assume that variables represent nonnegative real numbers. √125x^2

Verified step by step guidance

1. Recognize that the expression $\sqrt{125x^2}$ can be rewritten using the property of square roots: $\sqrt{a\cdot b}=\sqrt{a}\cdot \sqrt{b}$.
2. Decompose the number 125 into its prime factors: $125=5^3$.
3. Rewrite the expression as $\sqrt{5^3\cdot x^2}$.
4. Apply the property of square roots to separate the terms: $\sqrt{5^3}\cdot \sqrt{x^2}$.
5. Simplify each square root: $\sqrt{5^3}=5^{3/2}$ and $\sqrt{x^2}=x$. Combine these to express the simplified form.

Key Concepts

Product Rule

The product rule is a fundamental principle in calculus used to differentiate products of functions. It states that if you have two functions, u(x) and v(x), the derivative of their product is given by u'v + uv'. This rule is essential for simplifying expressions involving products, especially when combined with other algebraic operations.

Square Roots

A square root of a number is a value that, when multiplied by itself, gives the original number. For example, √x^2 equals x, assuming x is nonnegative. Understanding how to manipulate square roots is crucial for simplifying expressions, particularly when dealing with variables and constants in algebra.

Simplification of Expressions

Simplification involves rewriting an expression in a more manageable or understandable form without changing its value. This process often includes combining like terms, factoring, and applying algebraic rules such as the product rule. Mastery of simplification techniques is vital for solving algebraic problems efficiently.