Problem 90 - Elementary and Intermediate Algebra

Question

Simplify each expression, if possible. All variables represent positive real numbers. $$ 10 \sqrt[6]{12 x y}-\sqrt[6]{12 x y} $$

Step-by-Step Solution

Answer

The simplified expression is \( 9 \sqrt[6]{12xy} \).

Step 1: Identify Common Terms

The expression given is \( 10 \sqrt[6]{12x y} - \sqrt[6]{12 x y} \). Notice that both terms contain \( \sqrt[6]{12xy} \). This indicates that \( \sqrt[6]{12xy} \) is a common factor.

Step 2: Factor Out Common Term

Since \( \sqrt[6]{12xy} \) is common, we can factor it out: \[ (10 - 1) \cdot \sqrt[6]{12xy} \].

Step 3: Simplify the Expression Inside the Parentheses

Calculate the subtraction inside the parentheses: \( 10 - 1 = 9 \).

Step 4: Write the Simplified Expression

Now, replacing the subtraction result, the expression simplifies to: \( 9 \sqrt[6]{12xy} \).

Key Concepts

Radical Expressions

A radical expression involves roots, such as square roots or cube roots. In our exercise, we deal with sixth roots because we are looking at the expression \( \sqrt[6]{12xy} \). Root expressions help to simplify complex numbers and equations, especially when dealing with powers and exponents.
To simplify a radical expression, follow these steps:

• Identify any like radical terms, such as \( \sqrt[6]{12xy} \) in our example.
• Factor out the radicals if possible, as these can sometimes share common terms with others in the expression.
• Simplify further by performing arithmetic operations on coefficients.

Understanding radical expressions is crucial, especially as you progress into more advanced math courses. Radicals often appear in equations where you need to find real or complex solutions.

Factoring

Factoring helps break down larger expressions into more manageable parts. It's like finding pieces of a puzzle that fit together perfectly. In our example, the expression \( 10 \sqrt[6]{12xy} - \sqrt[6]{12xy} \) can be factored by recognizing the common term \( \sqrt[6]{12xy} \).

The fact that \( \sqrt[6]{12xy} \) appears in both terms allows us to factor it out of the expression. Factoring follows a process:

• Look for common factors in terms, much like in our example.
• Extract these factors and simplify the expression.
• Once factored, re-evaluate the expression to ensure it has been fully simplified.

Use this technique to simplify your problem-solving as you navigate algebraic expressions. Factoring turns a complex expression into a simpler, more comprehensible form.

Positive Real Numbers

"Positive real numbers" is a fundamental term in mathematics, referring to all the numbers greater than zero that can be expressed without using imaginary units. In algebra, you often deal with expressions that assume variables are positive real numbers. This assumption simplifies many operations.

• Positive numbers simply mean greater than zero.
• Real numbers include everything on the number line that's not an imaginary number.
• When solving or simplifying expressions, knowing numbers are positive avoids worrying about complex calculations involving these variables.

Working with positive real numbers allows you to apply various algebraic rules and properties confidently. In our exercise, assuming that the variables are positive ensures that the radical expressions result in real numbers.

Algebraic Expressions

Algebraic expressions contain numbers, variables, and operations. They're like sentences in algebra, used to describe quantities and relationships. Simplifying an algebraic expression means finding a simpler or more efficient way to write it, while still preserving its original meaning.

To simplify an algebraic expression, remember to:

• Combine like terms by looking for terms with the same variable and power. In our example, \( 10 \sqrt[6]{12xy} \) and \( - \sqrt[6]{12xy} \) are like terms.
• Use arithmetic operations like addition, subtraction, multiplication, and division.
• Factor out common terms where applicable, as shown in the exercise with \( \sqrt[6]{12xy} \).

By practicing simplification, you'll gain proficiency and confidence in managing algebraic tasks, equipping you for more advanced studies in math and science.