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Chapter 8: Problem 4

Simplify. $$1 \sqrt{v}+1 \sqrt{v}$$

Short Answer

Expert verified
The simplified expression is \(2 \sqrt{v} \).

Step by step solution

01

Identify the Terms

Recognize that you are dealing with two terms: \(1 \times \sqrt{v} \) and \(1 \times \sqrt{v} \)
02

Combine Like Terms

Since both terms are \( 1 \sqrt{v} \) and they are like terms, you can add the coefficients: \( 1 + 1 \).
03

Simplify the Coefficients

Add the coefficients together: \( 1 + 1 = 2 \).
04

Rewrite the Expression

Combine the simplified coefficient with the original term: \( 2 \sqrt{v} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

like terms
When simplifying expressions involving square roots, it's important to identify 'like terms'. Like terms are terms that contain the same variable raised to the same power.
For square roots, like terms will have the same radicand (the number or expression inside the square root).
This means that \(\sqrt{v}\) and \(\sqrt{v}\) are like terms.
Recognizing like terms allows us to combine them efficiently.
coefficients
Coefficients are the numbers in front of variables or square roots. In our exercise, the coefficients are both 1, as in \(\1 \sqrt{v}\) and \(\1 \sqrt{v}\).
When combining like terms with square roots, you add the coefficients together. It's like adding regular numbers but associated with the square root.
So, you add these coefficients: \(\1 + 1 = 2\). The resulting coefficient is then placed before the square root in the simplified expression.
square root addition
Adding square roots is straightforward when you are dealing with like terms. You can only combine them if the radicand (value under the square root) is the same.
In the exercise, since both terms are \(\sqrt{v}\), you can add them by adding their coefficients (1 and 1).
This results in the simplified form \(\2 \sqrt{v}\).
Remember that if the square roots are different (e.g., \(\sqrt{v}\) and \(\sqrt{w}\)), they cannot be directly added together. Only like terms can be combined in this manner.

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