Problem 457 - Elementary Algebra

Question

In the following exercises, simplify. (a) \(\sqrt[6]{r^{12}}\) (b) \(\sqrt[3]{s^{30}}\)

Step-by-Step Solution

Answer

Part (a): \(\ r^2\). Part (b): \(\ s^{10}\).

Step 1: Understand the Problem

The goal is to simplify the given radical expressions. For part (a), simplify \(\sqrt[6]{r^{12}}\). For part (b), simplify \(\sqrt[3]{s^{30}}\).

Step 2: Simplify the Expression in Part (a)

To simplify \(\sqrt[6]{r^{12}}\), recall that \(\sqrt[n]{x^m} = x^{m/n}\). Applying this rule, we get: \(\sqrt[6]{r^{12}} = r^{12/6}\). Simplify the exponent \(\frac{12}{6}\) to get \(\sqrt[6]{r^{12}} = r^2\).

Step 3: Simplify the Expression in Part (b)

To simplify \(\sqrt[3]{s^{30}}\), use the same rule: \(\sqrt[n]{x^m} = x^{m/n}\). Applying this rule, we get: \(\sqrt[3]{s^{30}} = s^{30/3}\). Simplify the exponent \(\frac{30}{3}\) to get \(\sqrt[3]{s^{30}} = s^{10}\).

Step 4: Write the Final Simplified Forms

The simplified form of \(\sqrt[6]{r^{12}}\) is \(\ r^2\), and the simplified form of \(\sqrt[3]{s^{30}}\) is \(\ s^{10}\).

Key Concepts

Exponent Rules

Exponent rules are essential when working with radical expressions. They help us to simplify complex expressions easily. An important rule to remember is the power rule for exponents:

• \( (x^a)^b = x^{a \times b} \).This rule states that when you raise an exponent to another exponent, you multiply the two exponents together.
  Another crucial rule is the quotient rule:
  
  • \( \frac{x^a}{x^b} = x^{a-b} \) This tells us that when you divide like bases, you subtract the exponents.
    Lastly, for radical simplification, we use:
    
    • \( \root n \rm x^m = x^{m/n} \) Meaning roots can be represented as fractional exponents, making them simpler to work with.
      Understanding these rules can make it much easier to simplify different kinds of expressions and radicals.

Radical Expressions

Radical expressions include square roots, cube roots, and higher-order roots. A radical expression looks like \( \root n \rm{x} \). The number \( n \) is the index, indicating the root level.

• For example, a square root is written as \( \root 2 \rm{x} \), often simplified to \( \rm{\backslashsqrt{x}} \), and a cube root as \( \root 3 \rm{x} \).To simplify radicals, we use the idea that an nth root can be expressed with exponents:
  The general form is
  
  • \( \root n \rm{x^m} = x^{m/n} \)By converting radicals into exponents, we can use our exponent rules to simplify them.
    In practice, consider the expression \( \root 6 \rm{r^{12}} \). We convert this into \( r^{12/6} \) and simplify the exponent to get \( r^2 \).This way, radical expressions become more manageable.

Simplification Steps

To simplify radical expressions, here are the steps to follow:

• Step 1: Convert the Radical to Exponential Form:
  Use \( \root n \rm x^m = x^{m/n} \) to convert any radical expression into a fraction exponent. For example, \( \root 3 \rm s^{30} \) becomes \( s^{30/3} \).Step 2: Simplify the Exponent:
  Perform the division in the exponent: \( s^{30/3} \) simplifies to \( s^{10} \). This involves basic arithmetic operations with the exponents.Step 3: Convert Back to Radical Form if Necessary:
  If the answer requires the form of a radical, you convert the fraction exponent back. Often, keeping it in simplified exponential form is clear and preferred.
  To sum up:
  • Start with conversion to exponential.
  • Simplify the exponent.
  • Convert back if needed.
  This method is systematic and keeps the process straightforward and manageable, ensuring that any intricate details are covered along the way.