Problem 65 - College Algebra with Modeling and Visualization

Question

Simplify the expression. Assume that all variables are positive. $$ \sqrt{x}+\sqrt{x}-\sqrt{y} $$

Step-by-Step Solution

Answer

The simplified expression is \( 2\sqrt{x} - \sqrt{y} \).

Step 1: Combine Like Terms

In the expression \( \sqrt{x} + \sqrt{x} - \sqrt{y} \), the first two terms \( \sqrt{x} + \sqrt{x} \) are like terms because they both contain \( \sqrt{x} \). Combine these like terms by adding them together: \( \sqrt{x} + \sqrt{x} = 2\sqrt{x} \). Now the expression becomes \( 2\sqrt{x} - \sqrt{y} \).

Step 2: Final Expression

After combining the like terms, you'll be left with the simplified expression \( 2\sqrt{x} - \sqrt{y} \). This is the simplest form because no further like terms can be combined or further simplifications made.

Key Concepts

Combining Like Terms

In algebra, terms that have identical variables and exponents are called 'like terms'. This means they can be combined together by performing the same operation on the numerical coefficients. In the given exercise, you are dealing with square roots, which can sometimes confuse students. However, the concept of like terms remains the same.

For example, if you have the terms \( \sqrt{x} + \sqrt{x} \), they are like terms because each includes the square root of \( x \). You can combine these terms by adding their coefficients, which in this case is the number 1 in front of each \( \sqrt{x} \). So, \( \sqrt{x} + \sqrt{x} = 2\sqrt{x} \).

• Identify terms with the same variable and root.
• Add or subtract numerical coefficients.
• Keep the variable and root the same.

When you combine like terms, you're simplifying expressions and making them easier to work with.

Square Roots

Square roots represent a value that, when multiplied by itself, gives the original number. In algebraic expressions, a square root is often denoted with the radical symbol \( \sqrt{} \). Understanding how square roots work is crucial when simplifying expressions that contain them.

In this exercise, the expression includes \( \sqrt{x} \) and \( \sqrt{y} \). Remember, square roots can only be combined directly when they are 'like' (i.e., \( \sqrt{x} + \sqrt{x} \) but not \( \sqrt{x} + \sqrt{y} \)). This is because combining like terms requires identical components under the radical.

• \( \sqrt{x \cdot y} = \sqrt{x} \cdot \sqrt{y} \)
• \( (\sqrt{x})^2 = x \)
• Combine square roots only when they have the same radicand.

Understanding these rules will help you manipulate square roots more effectively within algebraic expressions.

Algebraic Simplification

Algebraic simplification is the process of rewriting an expression in its simplest form. It involves reducing an expression by combining like terms and performing any operations indicated. Simplification helps in making expressions more manageable.

In the exercise, after combining like terms, we ended with the expression \( 2\sqrt{x} - \sqrt{y} \). This is considered simplified because there are no more like terms to combine. Algebraic simplification often relies on the order of operations and appropriate mathematical principles.

• Identify and combine like terms.
• Apply properties of square roots.
• Ensure there are no remaining operations to perform.

By simplifying an expression, you lay the groundwork for solving equations and inequalities, making finding solutions straight forward and efficient.