Problem 70 - Algebra for College Students

Question

Simplify each of the following. Express final results using positive exponents only. For example,\(\left(2 x^{\frac{1}{2}}\right)\left(3 x^{\frac{1}{3}}\right)=6 x^{\frac{5}{6}}\). \(\frac{18 x^{\frac{1}{2}}}{9 x^{\frac{1}{3}}}\)

Step-by-Step Solution

Answer

The simplified expression is \(2x^{\frac{1}{6}}\).

Step 1: Simplify the Coefficients

First, divide the numeric coefficients. The fraction given is \(\frac{18}{9}\). Divide 18 by 9 to get 2.

Step 2: Simplify the Exponents

Using the properties of exponents, subtract the exponent in the denominator from the exponent in the numerator for \(x\). The expression is \(x^{\frac{1}{2}}\) over \(x^{\frac{1}{3}}\). By the rule \(\frac{a^m}{a^n} = a^{m-n}\), this becomes \(x^{\frac{1}{2} - \frac{1}{3}}\).

Step 3: Calculate the Exponent

Calculate \(\frac{1}{2} - \frac{1}{3}\). To do this, find a common denominator. The least common denominator between 2 and 3 is 6. \(\frac{1}{2}\) is equivalent to \(\frac{3}{6}\), and \(\frac{1}{3}\) is equivalent to \(\frac{2}{6}\). Subtracting gives \(\frac{3}{6} - \frac{2}{6} = \frac{1}{6}\).

Step 4: Write the Final Expression

Combine the simplified coefficient with the exponent. The final expression is \(2x^{\frac{1}{6}}\).

Key Concepts

Properties of Exponents

Exponents are incredibly handy when simplifying algebraic expressions, as they offer various properties to streamline calculations. One of the core properties is the power of a quotient rule. This states that when you divide two exponents with the same base, you can subtract the exponents. More formally, this is written as \( \frac{a^m}{a^n} = a^{m-n} \). This rule helps simplify expressions like \( \frac{18 x^{\frac{1}{2}}}{9 x^{\frac{1}{3}}} \), where you handle the numeric coefficient separately from the variable, ensuring the final results retain simplicity.

Another useful property is the power of a power rule expressed as \( (a^m)^n = a^{m \cdot n} \). Although not directly used in the given solution, it's crucial in more complex problems. Understanding these properties can significantly reduce the effort required to work through algebraic expressions. Additionally, knowing how to convert between different formats, such as shifting between fractional exponents and roots, can put various concepts into perspective.

Positive Exponents

In mathematics, positive exponents represent straightforward calculations, typically indicating how many times to multiply a base by itself. Expressions formed with positive exponents often appear simpler and more manageable than their negative or fractional counterparts.

In problems like \( \frac{18 x^{\frac{1}{2}}}{9 x^{\frac{1}{3}}} \), converting any negative or fractional exponents to positive is a standard requirement. This makes the expression cleaner and easier to interpret. Positive exponents also align closely with the goal of finalizing answers in conventional, readable formats.

Working through exercises with exponents, especially when simplifying algebraic expressions, strengthens your grasp of mathematics and bolsters problem-solving skills. Simplifying to positive exponents involves recognition and practice, key to becoming proficient at algebra. In the exercise, we aim to simplify our results and achieve that all-important clarity by ensuring all exponents in the final answer are positive.

Fractional Exponents

Fractional exponents provide a unique way to express roots. The relationship is direct: a fractional exponent like \( x^{\frac{1}{2}} \) equates to the square root \( \sqrt{x} \), while \( x^{\frac{1}{3}} \) would correspond to the cube root \( \sqrt[3]{x} \). This reformulation often makes calculations involving roots simpler and more consistent with other exponent operations.

When you deal with fractional exponents in expressions, you can utilize the properties of exponents to consolidate or break down terms efficiently. For example, our expression \( \frac{18 x^{\frac{1}{2}}}{9 x^{\frac{1}{3}}} \) simplifies to \( 2 x^{\frac{1}{6}} \) by understanding that the subtraction of fractional exponents (\( \frac{1}{2} - \frac{1}{3} \)) results in another fractional exponent (\( \frac{1}{6} \)).

Mastering fractional exponents not only helps in algebraic simplification but provides a robust foundation for tackling calculus and more advanced areas. Engaging with these representations makes transitioning between different forms of expressions (such as shifting back to root forms, if needed) seamless and intuitive.