6.1.2: Homework

Reading Questions

1. What is the result of simplifying \(\sqrt[n]{a^n}\) when \(n\) is an even natural number? What if \(n\) is an odd natural number?
2. When simplifying \(\sqrt{x^6}\), why is the result \(|x^3|\) and not just \(x^3\) (unless \(x \ge 0\))?
3. What is the conjugate of the expression \(\sqrt{u} - \sqrt{6}\)?
4. When rationalizing a denominator (or numerator) that is a binomial involving a square root, what do you multiply the numerator and denominator by?
5. What is the product of an expression like \((a-b)\) and its conjugate \((a+b)\)?
6. When simplifying a difference quotient for a radical function like \(f(x) = \frac{1}{\sqrt{x}}\), what is the primary algebraic technique used to eliminate the radicals from the numerator after initial simplification?
7. In the context of simplifying difference quotients involving compound fractions with radicals, when should you distribute terms in the denominator, and when should you avoid it?
8. Simplify \(\sqrt[4]{(2q^3)^4}\), assuming \(q\) can be any real number.
9. Why is it important to rationalize the numerator when working with difference quotients of radical functions in Calculus?
10. If you are simplifying \(\frac{\sqrt{x+h} - \sqrt{x}}{h}\) by multiplying the numerator and denominator by \(\sqrt{x+h} + \sqrt{x}\), which part (numerator or denominator) should you fully multiply out (distribute), and which part should you generally leave in factored form? Why?

Homework

A textbook homework assignment needs to be created for this section.