Problem 94 - Algebra and Trigonometry

Question

Comparing Roots Without using a calculator, determine which number is larger in each pair. $$ \begin{array}{ll}{\text { (a) } 2^{1 / 2} \text { or } 2^{1 / 3}} & {\text { (b) }\left(\frac{1}{2}\right)^{1 / 2} \text { or }\left(\frac{1}{2}\right)^{1 / 3}} \\ {\text { (c) } 7^{1 / 4} \text { or } 4^{1 / 3}} & {\text { (d) } \sqrt[3]{5} \text { or } \sqrt{3}}\end{array} $$

Step-by-Step Solution

Answer

(a) \(2^{1/2}\), (b) \((1/2)^{1/3}\), (c) \(7^{1/4}\), (d) \(\sqrt{3}\).

Step 1: Compare Exponents - Part (a)

For Part (a), compare the exponents of 2 in \(2^{1/2}\) and \(2^{1/3}\). To do that, find a common denominator: \(1/2 = 3/6\) and \(1/3 = 2/6\). Since \(3/6 > 2/6\), \(2^{1/2}\) is larger than \(2^{1/3}\).

Step 2: Compare Exponents - Part (b)

In Part (b), compare the exponents for \((1/2)^{1/2}\) and \((1/2)^{1/3}\). Again, convert them using a common denominator: \(1/2 = 3/6\) and \(1/3 = 2/6\). Since \(3/6 > 2/6\), \((1/2)^{1/3}\) is larger because a larger exponent makes a smaller base's power smaller.

Step 3: Rationalizing Exponents - Part (c)

For Part (c), compare \(7^{1/4}\) and \(4^{1/3}\). Rewrite with a common base comparison, or test rationalizing their exponents. Compare their powers: \([7^{3/12}]\) and \([4^{4/12}]\). Since \(49/8 > 16/1\), \(7^{1/4}\) is larger.

Step 4: Rationalizing Exponents - Part (d)

Lastly, in Part (d), convert \(\sqrt[3]{5} = 5^{1/3}\) and \(\sqrt{3} = 3^{1/2}\) to the same base comparison. To compare, convert \(5^{2/6}\) and \(3^{3/6}\) as equivalent exponents. Since \(9 > 25\), \(3^{1/2}\) is larger.

Key Concepts

Comparing Exponents

When comparing exponents, it's crucial to analyze not only the base but also the fractional exponents themselves. This often involves comparing the exponents' sizes by identifying a common denominator.
For instance, if we look at the problem involving

• comparing \(2^{1/2}\) and \(2^{1/3}\), converting the exponents to a common base such as \(3/6\) and \(2/6\), clarifies that \(2^{1/2}\) is the larger because \(3/6\) is greater than \(2/6\).
• On the other hand, when comparing fractional bases like \((1/2)^{1/2}\) and \((1/2)^{1/3}\), it's important to note that a smaller base with a larger exponent can actually lead to a smaller overall number. Here, converting the exponents to the same denominators also resulted in \(1/3\) being larger compared to \(1/2\) in terms of base size.

Roots and Radicals

Roots and radicals are ways to represent fractional exponents. For instance, the square root of a number, like \( \sqrt{a} \), is essentially \(a^{1/2}\), and similarly for the cube root, \(\sqrt[3]{a} = a^{1/3}\).
This can be particularly helpful when trying to compare some bases.

• Take part (c) where comparing \(7^{1/4}\) and \(4^{1/3}\) might seem tricky at first. By changing them into a single denominator exponent, like converting them into \([7^{3/12}]\) and \([4^{4/12}]\), the comparison becomes easier. Here, interpreting roots as fractional exponents helps simplify and understand which is greater.
• Similarly, in part (d), converting \(\sqrt[3]{5}\) and \(\sqrt{3}\) directly into \(5^{1/3}\) and \(3^{1/2}\) surfaces the true comparison with base exponents, helping identify \(3^{1/2}\) as the larger value.

Rationalizing Exponents

Rationalizing exponents is a method used to make the comparison between numbers easier by ensuring the fractional exponents have a common denominator. Let's explore how this works with some examples:

• For example, in comparing \(7^{1/4}\) with \(4^{1/3}\), by expressing them as \(7^{3/12}\) and \(4^{4/12}\), you can easily see which of the terms has a greater value in relation to their respective powers. Here, comparing these forms shows \(7^{1/4}\) is greater because it has a higher effective power fraction.
• Similarly, in part (d), transforming \(5^{1/3}\) and \(3^{1/2}\) to the equivalent expressions \(5^{2/6}\) and \(3^{3/6}\) makes it evident that \(3^{1/2}\) (equivalent to \(3^{3/6}\)) is larger than \(5^{1/3}\) when analyzed on a consistent denominator basis.

This technique illuminates the relationship between base numbers and their exponents, allowing a clearer and direct pathway to organizing and comparing values even when the exponents are fractional.