Problem 492 - Elementary Algebra

Question

In the following exercises, simplify. (a) \(3 \sqrt[5]{9 x}+7 \sqrt[5]{9 x}\) \(8 \sqrt[7]{3 q}-2 \sqrt[7]{3 q}\)

Step-by-Step Solution

Answer

For (a), \(10 \sqrt[5]{9 x}\); for (b), \(6 \sqrt[7]{3 q}\).

Step 1: Identify like terms

Both terms in each expression must be of the same radical form to be combined. In this case, the terms \(3 \sqrt[5]{9 x}\) and \(7 \sqrt[5]{9 x}\), as well as \(8 \sqrt[7]{3 q}\) and \(2 \sqrt[7]{3 q}\), are like terms because they have the same radical index and radicand.

Step 2: Combine coefficients for the expression in (a)

For the expression \(3 \sqrt[5]{9 x} + 7 \sqrt[5]{9 x}\), add the coefficients 3 and 7: \(3 + 7 = 10 \). Thus, \(3 \sqrt[5]{9 x} + 7 \sqrt[5]{9 x} = 10 \sqrt[5]{9 x}\).

Step 3: Combine coefficients for the expression in (b)

For the expression \(8 \sqrt[7]{3 q} - 2 \sqrt[7]{3 q}\), subtract the coefficients 2 from 8: \(8 - 2 = 6\). Thus, \(8 \sqrt[7]{3 q} - 2 \sqrt[7]{3 q} = 6 \sqrt[7]{3 q}\).

Key Concepts

Like terms

When simplifying radical expressions, the first step is to identify 'like terms'.
Like terms in radical expressions mean that the terms have identical radical components—both the index and the radicand must be the same.
For example, in the expression \(3 \sqrt[5]{9x} + 7 \sqrt[5]{9x}\), both terms have the same radical index of 5 and the same radicand of \(9x\).
This makes them like terms and now you can combine them easily. Just remember that if either the radical index or the radicand is different, the terms aren't like terms and cannot be combined.
For instance, \(3 \sqrt[3]{5y}\) and \(7 \sqrt[5]{2z}\) are not like terms because their radicals are entirely different.

Combining coefficients

Once you identify like terms, the next step is combining their coefficients.
The coefficients are the numerical values that precede the radical expressions. For simplifying like terms, you only need to work with these coefficients.
Let's break it down:

• Take the expression \(3 \sqrt[5]{9x} + 7 \sqrt[5]{9x}\). Both terms have the same radical component, so focus on the coefficients 3 and 7.
  Add these coefficients: \(3 + 7 = 10\)
• Now, attach this sum to the radical part: \(10 \sqrt[5]{9x}\)

You can follow the same method for subtraction as well.

• Take another example like \(8 \sqrt[7]{3q} - 2 \sqrt[7]{3q}\). Both terms are like terms.
  Now focus on their coefficients, 8 and 2.
  Subtract the second coefficient from the first: \(8 - 2 = 6\)
• Then attach the difference to the radical part: \(6 \sqrt[7]{3q}\)

Combining coefficients in such a way simplifies your expression without changing its value.

Radical form

Understanding the concept of a 'radical form' is crucial for dealing with radical expressions.
A radical expression consists of two main parts: the radicand and the radical index.
The radicand is the quantity inside the radical symbol, and the radical index is the small number written just outside the radical symbol, indicating the root. For example, in the radical expression \( \sqrt[5]{9x} \), 9x is the radicand and 5 is the radical index.
To simplify, the radical form must be consistent.
Both the index and radicand need to match for terms to be simplified together.
Always look at the entire radical expression to determine its similarity to others before combining like terms and their coefficients.