Problem 96 - Elementary and Intermediate Algebra

Question

Use rational exponents to simplify each radical. All variables represent positive real numbers. See Example 10 . $$ \sqrt[12]{13^{4}} $$

Step-by-Step Solution

Answer

The simplified form of \( \sqrt[12]{13^{4}} \) is \( 13^{1/3} \).

Step 1: Convert the Radical to a Rational Exponent

The expression given is \( \sqrt[12]{13^{4}} \). This can be rewritten using rational exponents. A radical \( \sqrt[n]{a^m} \) can be expressed as \( a^{m/n} \). Therefore, \( \sqrt[12]{13^{4}} \) is the same as \( 13^{4/12} \).

Step 2: Simplify the Rational Exponent

Now, simplify the rational exponent \( \frac{4}{12} \). The greatest common divisor (GCD) of 4 and 12 is 4. Simplifying \( \frac{4}{12} \) by dividing both the numerator and the denominator by 4 gives \( \frac{1}{3} \). Thus, \( 13^{4/12} = 13^{1/3} \).

Step 3: Rewrite the Expression Using Simplified Exponent

Express the original problem in its simplified form using the simplified exponent. Hence, \( \sqrt[12]{13^{4}} \) simplifies to \( 13^{1/3} \).

Key Concepts

Simplifying Radicals

Simplifying radicals involves transforming a radical expression into a more manageable form. Radicals, such as square roots or cube roots, can often be expressed using rational exponents, which simplify the calculation process. When we use rational exponents, a radical expression like \( \sqrt[n]{a^m} \) is rewritten as \( a^{m/n} \). This allows us to simplify the powers and manipulate the expressions more easily.

• Identify the radical expression given.
• Convert the radical expression to a rational exponent following the form \( a^{m/n} \).
• Simplify the rational exponent if possible, by finding the greatest common divisor of the exponent's numerator and denominator.

This process makes it easier to handle and compare radical expressions, especially in algebraic operations. Simplifying radicals using rational exponents is a powerful tool in mathematics that makes complex calculations more accessible.

Radical Expressions

Radical expressions include any mathematical expression containing a radical sign (\(\sqrt{}\)). A radical expression often denotes the root of a number or variable. For example, \( \sqrt{16} = 4 \), because 4 squared is 16. Radical expressions can vary in complexity from basic square roots to higher-order roots like cube roots or fourth roots. They often appear in problems related to algebra and calculus.

• They typically include a radicand (the number under the root sign) and the degree of the root (indicated by the small number outside the radical)
• Radical expressions can be simplified by converting them to fractional powers, using rational exponents
• After conversion, expressions can be simplified further by finding common factors or using algebraic techniques

Understanding radical expressions is crucial for solving problems in both pure and applied mathematics. Recognizing how to manipulate and simplify them using rational exponents can greatly facilitate mathematical problem-solving.

Greatest Common Divisor

The greatest common divisor (GCD) is an essential concept when simplifying fractions or rational exponents. It is the largest integer that divides two numbers without leaving a remainder. When working with rational exponents, the numerator and the denominator of the fraction representing the exponent can often be reduced by the GCD.

• Find the factors of both the numerator and the denominator.
• Determine the largest number that appears in both sets of factors—this is the GCD.
• Simplify the fraction by dividing both the numerator and the denominator by the GCD.

For example, if the rational exponent is \( \frac{4}{12} \), the GCD of 4 and 12 is 4. By dividing both the numerator and the denominator by 4, you simplify \( \frac{4}{12} \) to \( \frac{1}{3} \). Utilizing the GCD is a reliable way to reduce fractions and exponents, making expressions easier to work with in mathematical calculations.