Problem 58 - A Graphical Approach to College Algebra

Question

If possible, simplify each radical expression. Assume that all variables represent positive real numbers. $$\sqrt[4]{\sqrt[3]{2}}$$

Step-by-Step Solution

Answer

Simplified expression: \(\sqrt[12]{2}\).

Step 1: Express Radicals in Fractional Exponents

Convert each radical expression into its equivalent form with fractional exponents. Start with the inner radical.\(\sqrt[3]{2}\) is expressed as \(2^{1/3}\).

Step 2: Apply the Outer Radical

Now consider the outer fourth root and apply it to the expression. The fourth root of an expression is equivalent to raising it to the power of \(1/4\).\(\sqrt[4]{2^{1/3}} = (2^{1/3})^{1/4}\).

Step 3: Simplify the Exponent

Combine the fractional exponents using the property \((a^m)^n = a^{m\cdot n}\).\((2^{1/3})^{1/4} = 2^{(1/3) \cdot (1/4)} = 2^{1/12}\).

Step 4: Rewrite in Radical Form

Convert the expression with a fractional exponent back into radical form. The exponent \(1/12\) translates to the twelfth root.\(2^{1/12} = \sqrt[12]{2}\).

Key Concepts

Fractional Exponents

Fractional exponents are another way to express roots and powers in mathematics. They provide a compact form of writing expressions that involve roots, like square roots or cube roots. When dealing with fractional exponents, the numerator typically represents the power, while the denominator indicates the root. For instance, if we have the expression

• \(x^{m/n}\)

This means we take the n-th root of \(x\) and raise it to the power of \(m\). Here's a quick example:

• \(x^{1/2}\) is the square root of \(x\) which can be written as \(\sqrt{x}\).
• \(x^{3/4}\) would mean take the fourth root of \(x\), and then raise it to the third power.

Using fractional exponents simplifies arithmetic operations with radicals, making solving and simplifying expressions easier and more straightforward.

Radical Form Conversion

Converting between radical form and fractional exponents is a fundamental skill in simplifying expressions. This conversion is crucial when you encounter nested radicals or when you want to use exponent rules to simplify expressions.The process involves representing a root, such as a square root \(\sqrt{x}\) or cube root \(\sqrt[3]{x}\), as an exponent. For instance:

• The expression \(\sqrt[3]{x}\) in radical form is equivalent to \(x^{1/3}\) if written with fractional exponents.

When converting back to radical form from fractional exponents:

• Identify the denominator of the exponent, as this represents the root.
• Match it to the appropriate radical, such as using the fifth root \(\sqrt[5]{y}\) for exponents of \(y^{1/5}\).

Converting expressions allows you to change the format of your problem to one that may be more manageable for simplification or further calculation.

Exponent Properties

Exponent properties are a set of rules that make it easier to work with exponential expressions. They help simplify problems and allow us to manipulate expressions for easier computation. Here are some key properties of exponents that are especially useful:1. **Power of a Power**: - When you have an exponent raised to another power, such as \((a^m)^n\), you multiply the exponents together to get \(a^{m \cdot n}\).2. **Product of Powers**: - When multiplying like bases, you add the exponents, like \(a^m \times a^n = a^{m+n}\).3. **Quotient of Powers**: - When dividing like bases, subtract the exponents: \(a^m / a^n = a^{m-n}\).4. **Zero Exponent**: - Any base raised to the zero power is 1: \(a^0 = 1\) provided \(a eq 0\).These properties allow us to transform complex expressions into simpler forms, making it easier to solve equations and inequalities. They are extremely handy when manipulating expressions with fractional exponents or converting between radical and exponential forms.