Problem 87 - College Algebra and Calculus: An Applied Approach

Question

Complete the statement with \(<,=\), or \(\geq$$\sqrt{3} \cdot \sqrt[4]{3} \quad \sqrt[8]{3}\)

Step-by-Step Solution

Answer

\(\sqrt{3} \cdot \sqrt[4]{3} > \sqrt[8]{3}\).

Step 1: Convert Square Roots to Power of 3

Convert \(\sqrt{3}\) to 3's power, we have \(3^{1/2}\). Convert \(\sqrt[4]{3}\) to 3's power, it becomes \(3^{1/4}\).

Step 2: Combine the Two Power Terms

Now, multiply the two terms: \(3^{1/2} \cdot 3^{1/4} = 3^{1/2+1/4} = 3^{3/4}\).

Step 3: Convert the Third Root to Power of 3

Similarly, convert \(\sqrt[8]{3}\) to 3's power, we get \(3^{1/8}\).

Step 4: Compare the Two Power Terms

Now we have to compare \(3^{3/4}\) and \(3^{1/8}\). Since 3/4 is greater than 1/8, we can conclude that \(3^{3/4} > 3^{1/8}\).

Key Concepts

Square Roots to Exponents

Understanding how to convert square roots to exponents is crucial for simplifying expressions and solving equations involving radicals. A square root, such as \(\sqrt{x}\), can be expressed as an exponent by recognizing it as \(x^{1/2}\). This follows from the definition that the square root of a number is a value that, when multiplied by itself, gives the original number. In other words, \(\sqrt{x} \cdot \sqrt{x} = x\), which is the same as \(x^{1/2} \cdot x^{1/2} = x^{1/2+1/2} = x^1\).

By converting square roots to exponents, we can apply the rules of exponents to simplify and manipulate expressions. This skill is particularly handy when you need to compare radical expressions, involve radicals in your calculations, or solve complex equations.

Multiplying Powers with the Same Base

When you're faced with multiplying powers that have the same base, there's a simple rule to follow: add the exponents. If you have \(a^m \cdot a^n\), the result is \(a^{m+n}\). This rule applies because when you multiply two powers of the same base, you're essentially multiplying the base by itself many times.

Here's a visual way to understand it: if you have \(a \cdot a \cdot a\) (which is \(a^3\)) and you multiply it by \(a \cdot a\) (which is \(a^2\)), you'll end up with five a's being multiplied together, which gives you \(a^5\). Hence, \(a^3 \cdot a^2 = a^{3+2} = a^5\). This concept is fundamental when simplifying expressions involving variables raised to powers, and it helps ensure that your work stays neat and manageable.

Radical and Exponent Conversion

Radicals and exponents are two sides of the same coin. The process of converting between them allows for easier comparison and manipulation of expressions. Any radical with index \(n\), like \(\sqrt[n]{x}\), can be written as an exponent by using the fraction \(1/n\), resulting in \(x^{1/n}\).

Conversely, you can convert an expression with a fractional exponent back to radical form. For example, \(x^{3/4}\) can be written as \(\sqrt[4]{x^3}\). This interchangeability is incredibly useful for performing operations that are more complex with radicals than with exponents. Understanding this concept will empower you to tackle a wide range of algebraic problems more effectively and confidently.

Comparing Powers of the Same Base

Comparing powers of the same base can be straightforward when you know how to handle the exponents. Given two expressions like \(a^m\) and \(a^n\), where \(a\) is the base and \(m\) and \(n\) are the exponents, it's the size of the exponents that will determine which expression is greater.

If the base \(a\) is greater than 1 and \(m > n\), then \(a^m > a^n\). Conversely, if \(m < n\), then \(a^m < a^n\). This rule remains true regardless of whether the exponents are whole numbers, fractions, or even negative numbers. Mastery of this concept will serve as a cornerstone for algebra, helping students understand growth rates, solve inequalities, and grasp the behavior of exponential functions.