Problem 101 - Elementary and Intermediate Algebra

Question

Use rational exponents to simplify each radical. All variables represent positive real numbers. See Example 10 . $$ \sqrt[9]{\sqrt{c}} $$

Step-by-Step Solution

Answer

The simplified expression is \( c^{1/18} \).

Step 1: Express Each Radical Using Rational Exponents

The expression given is \( \sqrt[9]{\sqrt{c}} \). Recall that a square root \( \sqrt{c} \) can be expressed with a rational exponent as \( c^{1/2} \). Similarly, a ninth root is written with an exponent of \( 1/9 \). Hence, the expression \( \sqrt[9]{\sqrt{c}} \) becomes \( (c^{1/2})^{1/9} \) in terms of rational exponents.

Step 2: Apply the Power of a Power Rule

Using the rule \( (a^m)^n = a^{m \cdot n} \), we simplify \( (c^{1/2})^{1/9} \). Multiply the exponents: \( 1/2 \times 1/9 = 1/18 \). So, the expression simplifies to \( c^{1/18} \).

Step 3: Restate the Simplified Expression

The original expression \( \sqrt[9]{\sqrt{c}} \) is now simplified to \( c^{1/18} \) using rational exponents.

Key Concepts

Simplifying Radicals

When we talk about simplifying radicals, we're really looking at ways to rewrite expressions involving roots in a simpler or more useful form. In mathematics, a radical typically refers to a root, such as square roots or cube roots. You're probably familiar with the square root sign (√) that indicates the principal square root of a number. In our case, simplifying radicals means converting them into expressions with rational exponents. This process has several benefits:

• It often makes complex expressions easier to handle mathematically.
• Allows the use of exponent laws which are sometimes more straightforward to apply.
• Can be vital in various fields of mathematics and science where precise calculations are necessary.

For example, the expression \( \sqrt[9]{\sqrt{c}} \) can be rewritten as a combination of exponents. The square root \( \sqrt{c} \) translates to \( c^{1/2} \), and taking the ninth root afterward gives us \( (c^{1/2})^{1/9} \). This conversion opens the door to other methods of simplification using exponent rules.

Power of a Power Rule

One crucial exponent law that helps in simplifying expressions with rational exponents is the power of a power rule. This rule states: \[(a^m)^n = a^{m \cdot n}\]This might look a bit complex, but it's essential for making formulas easier to work with. When you raise a power to another power, you multiply the exponents. Let's see how this works with our example:

• Starting with the expression \( (c^{1/2})^{1/9} \).
• Using the power of a power rule, multiply the exponents \( 1/2 \times 1/9 \).
• This gives \( 1/18 \).

The expression \( (c^{1/2})^{1/9} \) thus simplifies to \( c^{1/18} \) using this straightforward application of the power of a power rule. In mathematics, applying these laws helps maintain accuracy and simplicity in calculations.

Rational Expressions

Rational expressions use rational exponents, which means numbers that are expressed as fractions can act as exponents. When dealing with rational expressions, you're frequently turning roots into fractional powers. This is incredibly useful for solving equations, simplifying expressions, and making the math more manageable. Take a closer look at our example:

• Initially, the nested radicals \( \sqrt[9]{\sqrt{c}} \) involve a square root and a ninth root.
• Express these as \( c^{1/2} \) and \((c^{1/2})^{1/9} \), respectively.
• After applying the power of a power rule, it's simplified to \( c^{1/18} \).

Why bother with building rational expressions? Because it:

• Facilitates the application of exponent rules.
• Makes calculations more precise.
• Is essential in higher-level math, such as calculus and beyond.

Thus, converting to a rational expression strategically simplifies complex operations and calculations.