Question

Simplify each expression. Assume that all variables are positive when they appear. $$3 \sqrt{2}+4 \sqrt{2}$$

Short answer

7 \sqrt{2}\

Step-by-step solution

- Identify Like Terms

We need to check if the terms under the square roots are the same. In this case, both terms are under \(\sqrt{2}\), which means they are like terms.

- Add the Coefficients

Since \(3 \sqrt{2} + 4 \sqrt{2}\) are like terms, we can add their coefficients just as we would with regular numbers. The coefficients here are 3 and 4.

- Combine the Terms

Adding the coefficients, we get: \(3+4 = 7\). Therefore, \(3 \sqrt{2} + 4 \sqrt{2} = 7 \sqrt{2}\).

Key concepts

like terms

Understanding 'like terms' is key when simplifying radical expressions.
Like terms are terms that have the same variable parts, including the same exponents and radicands (the number under the root).
For instance, in the expression 3√2 + 4√2, both terms are like terms because they both have the same radicand of √2.
Like terms can be combined by adding or subtracting their coefficients. This is similar to adding numbers with the same variable in algebra (e.g., 3x + 4x = 7x).
All you need to do is focus on the numerical coefficients and factor out the common variable or radical part.
This results in a single term that represents the summation or subtraction of the quantities. In our example, 3 and 4 are coefficients, and they add up to 7 while keeping the radicand (√2) unchanged.

coefficients

In radical expressions, coefficients are the numbers placed in front of the radicals. They play a crucial role in the addition or subtraction of like terms.
Consider the expression 3√2 + 4√2:

• The coefficient 3 is in front of √2 in the first term.
• The coefficient 4 is in front of √2 in the second term.

Since the terms are like terms (because of the same √2), we can add the coefficients directly. The arithmetic operation is akin to adding plain numbers:
3 + 4 = 7.
The result is then written as 7√2. This demonstrates how straightforward it is to manipulate coefficients to simplify expressions when dealing with like radicals.
Remember, the radicand must be identical for this simplification method to apply.

addition of radicals

The addition of radicals follows rules similar to those of adding algebraic expressions. Here, you need like terms – in other words, radicals that have the same radicand.
For example, consider: 3√2 + 4√2. Here are the steps:
First, check if the radicals have the same radicand (which is √2 in this case).
Since they do, you move on to the coefficients. This process is akin to adding fractions with a common denominator.
You just add the coefficients: 3 + 4, and then attach the common radicand (√2) to the result.
So, 3√2 + 4√2 simplifies to 7√2.
This method helps to streamline and simplify expressions involving radicals, making it much easier to deal with them in algebraic equations or further mathematical operations.