Problem 71 - Algebra A Combined Function

Question

Use rational exponents to simplify each radical. Assume that all variables represent positive real numbers. $$ \sqrt[12]{a^{8} b^{4}} $$

Step-by-Step Solution

Answer

The simplified form is \( a^{\frac{2}{3}} b^{\frac{1}{3}} \).

Step 1: Express the Radicals using Rational Exponents

First, rewrite the expression \( \sqrt[12]{a^8 b^4} \) using rational exponents. Radical expressions can be expressed as a fraction power, where the numerator is the exponent and the denominator is the root. Thus, \( \sqrt[12]{a^8} \) becomes \( a^{\frac{8}{12}} \) and \( \sqrt[12]{b^4} \) becomes \( b^{\frac{4}{12}} \). Combine them: \[ (a^8 b^4)^{\frac{1}{12}} = a^{\frac{8}{12}} b^{\frac{4}{12}} \].

Step 2: Simplify the Exponents

Simplify the rational exponents. Both \( \frac{8}{12} \) and \( \frac{4}{12} \) can be reduced by dividing the numerator and the denominator by their greatest common divisor. For \( \frac{8}{12} \), divide both terms by 4 to get \( \frac{2}{3} \).For \( \frac{4}{12} \), divide both terms by 4 to get \( \frac{1}{3} \).Thus, the expression becomes: \[ a^{\frac{2}{3}} b^{\frac{1}{3}} \].

Step 3: Combine and Simplify the Expression

Now, combine the simplified exponents: Since there is nothing further to combine or simplify, the expression \( a^{\frac{2}{3}} b^{\frac{1}{3}} \) is the simplest form.

Key Concepts

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, and higher-order roots. Typically, these are represented with a radical sign (√) and an index that tells us which root to take, e.g., the fourth, fifth, or twelfth root. In the expression \( \sqrt[12]{a^8 b^4} \), the index is 12, indicating we need the twelfth root.

To better understand radical expressions, it helps to break them down:

• The radical symbol "\( \sqrt{} \)" indicates a root operation.
• The number outside the radical, called the index, shows the degree of the root.
• Inside the operand are the sub-expressions or numbers we're operating on - here, \( a^8 \) and \( b^4 \).

Radical expressions can seem complex, but starting by converting them into rational exponents, or fraction form, simplifies the process. By doing this, we make use of exponent rules to manipulate the expression more easily.

Simplifying Expressions

Simplifying expressions is about reducing a given expression to its simplest form without changing its value. This process is critical, especially in algebra, as it helps in making calculations more straightforward. In the case of radical expressions, simplifying often involves using properties of exponents:

• Finding a common factor for the terms involved.
• Reducing fractions to lower terms.
• Using properties of exponents to combine or separate terms.

In our example exercise, once we convert radicals into rational exponents, we simplify the fractions by dividing both the numerator and the denominator by their greatest common divisor (GCD). For instance:

• For \( \frac{8}{12} \), both 8 and 12 can be divided by 4, simplifying to \( \frac{2}{3} \).
• Similarly, \( \frac{4}{12} \) simplifies to \( \frac{1}{3} \).

This simplification makes the expression easier to evaluate or further manipulate in problem-solving scenarios.

Rational Exponents to Radicals

Transforming rational exponents to radicals, and vice versa, is a crucial skill in algebra. Understanding how these relate is vital for simplifying and evaluating expressions. The primary concept to grasp is that a rational exponent \( \frac{m}{n} \) is equivalent to taking the nth root of a base raised to the mth power.

Consider this:

• The expression \( x^{\frac{m}{n}} \) can be rewritten as \( \sqrt[n]{x^m} \), where n is the root's index, and m is the power to which the base x is raised.
• Alternatively, you can write it as \( (\sqrt[n]{x})^m \), choosing whichever form makes your calculation more straightforward.

In our exercise, we take the terms \( a^8 \) and \( b^4 \) under a 12th root and convert them into \( a^{\frac{8}{12}} \) and \( b^{\frac{4}{12}} \), respectively. This conversion allows us to more easily apply exponent rules to simplify the expression.

Positive Real Numbers

When working with radical expressions and rational exponents, it's important to consider the nature of the numbers involved. In mathematics problems, especially those involving roots, we frequently assume that the variables represent positive real numbers.

This assumption helps in simplifying expressions and avoiding complex numbers. Key considerations include:

• Roots of positive numbers remain real; for example, the square root or any even root of a positive number is always positive.
• Assuming variables as positive ensures all steps, like simplification and conversion, are valid without having to deal with undefined operations or imaginary numbers.

In the given exercise, stating that variables represent positive real numbers allows us to seamlessly transition from radicals to rational exponents and simplify them without concern for complex roots or additional restrictions. This assumption keeps our focus on learning exponent rules and simplification techniques.