Problem 18 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

If the statement is always true, write "True;" if the statement is not always true, produce a counterexample. In these problems, \(a\) and \(b\) are positive constants. $$ \sqrt{\frac{a^{2 x}}{b^{-2 x}}}=(a b)^{x} $$

Step-by-Step Solution

Answer

True

Step 1: Simplify the left-hand side (LHS)

Consider \(\sqrt{\frac{a^{2 x}}{b^{-2 x}}}\). This can be simplified using the properties of exponents, remembering that \(b^{-2x} = 1/b^{2x}\), and thus it simplifies to \( \sqrt{a^{2x} * b^{2x}} \)

Step 2: Take the square root

Taking the square root of the expression, \( \sqrt{a^{2x} * b^{2x}} \) simplifies to \( (ab)^x \) using the properties of radicals, as \( \sqrt{x^2} = x \).

Step 3: Compare left-hand side with right-hand side

After simplifying the left-hand side, one gets \( (ab)^x \), which is the same as the right-hand side of the equation.

Key Concepts

Simplifying Radical Expressions

Radical expressions often appear intimidating, but they follow specific rules that, once mastered, make simplifying them straightforward. A radical expression is an expression that includes a radical symbol, typically used to denote the square root or other roots.

To simplify a radical expression, one should look for factors of the expression that are perfect squares (or perfect cubes for cube roots, etc.), and then take them out of the radical. For instance, \( \sqrt{a^{2} * b^{2}} \) simplifies to \( ab \) because \( a^{2} \) and \( b^{2} \) are both perfect squares. Additional steps include rationalizing the denominator if necessary, which means eliminating the radical from the denominator of a fraction. By practicing these techniques, complex radical expressions can often be rewritten in a much simpler form.

Properties of Radicals

Understanding the properties of radicals is the cornerstone of effectively simplifying radical expressions. Among the essential properties is the fact that the \(n\)th root of a product equals the product of the \(n\)th roots of each factor separately, meaning \( \sqrt[n]{a*b} = \sqrt[n]{a} * \sqrt[n]{b} \).

Another significant property is that the \(n\)th root of a quotient is the quotient of the \(n\)th roots: \( \sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}} \). When dealing with radicals, it is also essential to remember that \( \sqrt[a]{b^{a}} = b \) provided that \(b\) is positive when \(a\) is even, as the root and the power 'cancel' each other. This property plays a crucial role in simplifying radical expressions and is often used in algebraic proofs.

Proofs in Algebra

Proofs in algebra involve demonstrating that a particular algebraic statement is true for all allowed values by using logical reasoning and established algebraic rules. Algebraic proofs require a thorough understanding of concepts like properties of exponents and radicals.

For the given exercise, the initial equality \( \sqrt{\frac{a^{2x}}{b^{-2x}}} = (ab)^{x} \) can be approached as an algebraic proof. By simplifying the left-hand side through known properties—converting the negative exponent to a positive and taking roots of exponential expressions—the proof concludes when both sides of the equation are shown to be equal. The critical takeaway in proofs is the logical sequence of steps and proper application of algebra rules to demonstrate validity.

Rational Exponents

Rational exponents are another way of expressing roots, providing a convenient notation for radical expressions. The expression \( a^{\frac{m}{n}} \) is equivalent to \( \sqrt[n]{a^{m}} \) or the \(n\)th root of \(a\) raised to the power of \(m\). This property facilitates the simplification of expressions and solving equations that would otherwise be cumbersome with traditional radical notation.

In the context of the exercise, \(a^{2x}\) and \(b^{-2x}\) can be interpreted as \(a\) and \(b\) being raised to rational exponents, ensuring a smooth transition between radicals and exponents. Understanding rational exponents allows a deeper comprehension of how powers and roots interact and how they obey consistent rules, regardless of the notation used.