Problem 146 - Elementary Algebra

Question

In the following exercises, simplify. $$ 7 \sqrt{2}-3 \sqrt{2} $$

Step-by-Step Solution

Answer

4 \(\sqrt{2}\)

Step 1: Identify Like Terms

Recognize that both terms, \(7 \sqrt{2}\) and \(-3 \sqrt{2}\), contain the same radicand, which is \(\sqrt{2}\). This implies that they are like terms and can be combined through addition or subtraction.

Step 2: Combine Like Terms

Subtract the coefficients of \( \sqrt{2} \) : \(7 - 3\). This yields: \(4 \sqrt{2}\)

Step 3: Simplify the Expression

The simplified form of the expression is \(4 \sqrt{2}\).

Key Concepts

like terms

When simplifying radical expressions, it's essential to recognize 'like terms'. These are terms that share the same variable part, including exponents and radicals. In the context of the exercise, \(7 \sqrt{2}\) and \ -3 \sqrt{2} \ are considered like terms because they both contain the same radical part, which is \(\sqrt{2}\). This similarity allows us to combine these terms through addition or subtraction.
Here's a simple analogy for better understanding: Think of 'like terms' as similar fruits, like apples. Combining 7 apples and 3 apples gives you a total of 10 apples. Similar logic applies to like terms with the same radical. We can combine them directly by adding or subtracting their coefficients, which are the numerical parts in front of the radicals.

radicand

Understanding the 'radicand' is crucial when dealing with radicals. The radicand is the number or expression inside the radical symbol (√) that you’re trying to find the root of. For instance, \( \sqrt{2} \) has a radicand of 2. In our exercise, both \( 7 \sqrt{2}\) and \ -3 \sqrt{2} \ share the same radicand, which enables them to be treated as like terms and combined.
The radicand determines the type of root you are dealing with. It’s important because it informs you whether the terms can be simplified together. If the radicands are different, the terms are not like terms and cannot be directly combined through addition or subtraction.

coefficients

In algebra, 'coefficients' refer to the numerical parts of terms that include variables or radicals. They are the leading numbers used to multiply the variables. In the exercise \ 7 \sqrt{2} - 3 \sqrt{2}\ , the coefficients are 7 and -3, respectively. Coefficients are crucial because they help in combining like terms.
In the provided solution, we saw how the coefficients were subtracted: \(7 - 3 = 4\). This results in the simplified expression: \(4 \sqrt{2}\). Essentially, understanding how to handle coefficients will enable you to simplify various expressions accurately and efficiently. Remember, only like terms with the same radicand can have their coefficients combined.