Problem 77 - Algebra for College Students

Question

Simplify each of the following. Express final results using positive exponents only. For example,\(\left(2 x^{\frac{1}{2}}\right)\left(3 x^{\frac{1}{3}}\right)=6 x^{\frac{5}{6}}\). \(\left(\frac{18 x^{\frac{1}{3}}}{9 x^{\frac{1}{4}}}\right)^{2}\)

Step-by-Step Solution

Answer

The simplified expression is \( 4x^{\frac{1}{6}} \).

Step 1: Simplify the Fraction Inside the Parentheses

The expression inside the parentheses is \( \frac{18 x^{\frac{1}{3}}}{9 x^{\frac{1}{4}}} \). Divide the coefficients: \( \frac{18}{9} = 2 \). For the exponents, use the property \( x^{a}/x^{b} = x^{a-b} \). So, divide \( x^{\frac{1}{3}} \) by \( x^{\frac{1}{4}} \), which results in \( x^{\frac{1}{3} - \frac{1}{4}} = x^{\frac{4}{12} - \frac{3}{12}} = x^{\frac{1}{12}} \). The fraction inside the parentheses simplifies to \( 2x^{\frac{1}{12}} \).

Step 2: Apply the Square to Simplify the Expression

Take the simplified expression \( 2x^{\frac{1}{12}} \) and apply the square: \( (2x^{\frac{1}{12}})^{2} \). Use the property \((ab)^n = a^n b^n\) to obtain \( 2^2 (x^{\frac{1}{12}})^2 \). Calculate the square of the coefficient: \( 2^2 = 4 \). For the exponent, multiply \( \frac{1}{12} \times 2 = \frac{2}{12} = \frac{1}{6} \). So the expression becomes \( 4x^{\frac{1}{6}} \).

Step 3: Ensure Positive Exponents

Check that all exponents in the expression \( 4x^{\frac{1}{6}} \) are positive. Since \( \frac{1}{6} \) is already a positive exponent, there’s no further simplification needed.

Key Concepts

Simplifying Expressions

Simplifying expressions is like tidying up a messy room: it involves combining and reducing terms to create a neater, more manageable result. When you're faced with expressions inside parentheses that include coefficients and variables with exponents, the goal is to simplify them step by step.
Here’s how you might do it:

• Identify the Components: Break down the expression into its parts, such as coefficients (numbers) and variables (like \(x\), \(y\)).
• Use Exponent Rules: Remember that multiplying exponents means adding powers, and dividing means subtracting them. This makes simplification straightforward—especially if you need to ensure every term's exponents are positive. For instance, \(x^{\frac{1}{3}}/x^{\frac{1}{4}} = x^{\frac{1}{3} - \frac{1}{4}}\).
• Combine Like Terms: Once exponents are managed, look for like terms, which are terms that can be combined together, such as same variable factors with modified exponents.

After executing these steps, you'll have a simpler form of the expression, which is easier to analyze or solve in subsequent calculations.

Fraction Simplification

Fraction simplification involves reducing fractions to their simplest form by finding common factors in the numerator and denominator. This process resembles simplifying any numerical or algebraic fraction.
Consider a fraction like \(\frac{18x^{\frac{1}{3}}}{9x^{\frac{1}{4}}}\). Here's how to simplify it:

• Divide the Coefficients: Simplify the numbers first by dividing the top coefficient by the bottom—\(\frac{18}{9} = 2\).
• Manage the Exponents: For the exponents, use the rule \(x^a / x^b = x^{a-b}\), which simplifies fractional exponents by subtracting the bottom's power from the top's power. This results in \(x^{\frac{1}{12}}\) after simplifying \(x^{\frac{1}{3}}/x^{\frac{1}{4}}\).

Reducing fractions in this systematic way makes complex expressions much simpler, setting the stage for further operations like exponentiation.

Positive Exponents

While simplifying expressions involving exponents, ensuring the result is expressed as a positive exponent is crucial for clarity and convention in mathematics. Positive exponents indicate straightforward multiplication, which is easier to conceptualize and work with.
Here’s how to ensure positive exponents:

• Convert any Negative Exponents: If an exponent is negative, recall that it can be transformed into a positive one by taking the reciprocal of the base: \(x^{-a} = \frac{1}{x^a}\).
• Operate with Fractions: Suppose you’ve simplified an expression and end with a fraction in the exponent, multiply through as shown: \((x^{\frac{1}{12}})^2 = x^{\frac{1}{6}}\). Ensure your final result entails only positive numbers.

Having your final expression in terms of positive exponents makes it consistent with general mathematical solutions, enhancing readability and further arithmetic operations.