Problem 90 - 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: Factor out the Common Term

Notice that both terms involve the 6th root of \(12xy\). We can factor out \(\sqrt[6]{12xy}\) from both terms.This gives us:\[10 \sqrt[6]{12xy} - \sqrt[6]{12xy} = \left(10 - 1\right) \sqrt[6]{12xy}\]

Step 2: Simplify the Coefficient

Subtract the coefficients inside the parentheses:\[10 - 1 = 9\]So the expression becomes:\[9 \sqrt[6]{12xy}\]​

Key Concepts

Factoring

Factoring is a crucial skill in algebra that helps simplify expressions and solve equations. It involves breaking down an expression into simpler components or 'factors' that, when multiplied, reproduce the original expression. In the problem given, when we look at the expression \(10 \sqrt[6]{12xy} - \sqrt[6]{12xy}\), we see a common term: \(\sqrt[6]{12xy}\).
This commonality allows us to factor it out of both expressions, much like factoring out a common factor from polynomial terms in simpler algebra.

• This leaves us with \(10 - 1\) inside the parentheses since both terms share \(\sqrt[6]{12xy}\) as a factor.
• The process is similar to factoring out a common variable or number in simpler expressions, like distinguishing common terms in \(2x + 4x = 2(x + 2x)\).

Once factored, the problem becomes much simpler to handle. This step is foundational in radical simplification, setting the stage for further simplification of coefficients.

Coefficients

In algebraic expressions, coefficients are the numerical or constant parts that multiply the variables or radical expressions. In our problem, the coefficients are the numbers that multiply the radical expression \(\sqrt[6]{12xy}\). Initially, our expression is \(10 \sqrt[6]{12xy} - 1 \sqrt[6]{12xy}\).
After factoring out the common radical, we found ourselves with a much simpler coefficient operation: \(10 - 1\).
This subtraction of coefficients results in a new coefficient, which in this case is 9.

• This simplification to \(9 \sqrt[6]{12xy}\) highlights how mathematical operations on coefficients can reduce the complexity of an expression.
• Coefficient simplification is an essential step in ensuring that expressions are as simplified as possible, reducing confusion and making further operations or comparisons much easier.

Variables in Roots

Understanding variables within radical expressions is key to simplifying such problems. When you encounter variables under a radical, whether a square root, cube root, or any other root like the sixth root here, it's essential to handle them carefully.
In this exercise, \(\sqrt[6]{12xy}\), the variable components \(x\) and \(y\) contribute to the complexity of the expression. Variables within a root add another layer of complexity because they are part of the radicand — the number or expression under the root symbol.

• The assumption that these variables represent positive real numbers is crucial because it means you don't have to worry about negative roots, which can fundamentally change the problem.
• In contexts where further simplification might occur, you would consider their prime factors or pairings, but here, the task is purely to factor out and combine based on a shared radical form.

Having a clear understanding of variables in roots allows for deeper insight into algebraic simplification and manipulation, especially useful in more advanced mathematics contexts.