Question

Challenge Problems. Perform the indicated operation and simplify. $$(\sqrt{x}+\sqrt{y}) \times(\sqrt{x}+\sqrt{y})$$

Short answer

The simplified form of the expression is \(x + 2\sqrt{xy} + y\).

Step-by-step solution

Apply the foil method

Use the FOIL (First, Outer, Inner, Last) method to multiply two binomials. Multiply First terms, Outer terms, Inner terms, and Last terms.

Multiply the First terms

Multiply the first terms in each binomial: \(\sqrt{x} \times \sqrt{x} = x\).

Multiply the Outer terms

Multiply the outer terms: \(\sqrt{x} \times \sqrt{y} = \sqrt{xy}\).

Multiply the Inner terms

Multiply the inner terms using commutative property of multiplication (which tells us that \(a \times b = b \times a\)): \(\sqrt{y} \times \sqrt{x} = \sqrt{yx} = \sqrt{xy}\).

Multiply the Last terms

Multiply the last terms: \(\sqrt{y} \times \sqrt{y} = y\).

Combine like terms

Combine the results of steps 2 to 5 to get the expression: \(x + 2\sqrt{xy} + y\).

Write the final simplified expression

The final simplified expression is \(x + 2\sqrt{xy} + y\), with like terms combined and no further simplification possible.

Key concepts

FOIL Method

The FOIL method stands for First, Outer, Inner, Last, and is a technique used to multiply two binomials. Binomials are algebraic expressions containing two terms, such as \(a + b\) and \(c + d\). To use the FOIL method, you multiply each term in the first binomial by each term in the second binomial in a specific order.

Let's apply the FOIL method to the given problem \( (\sqrt{x} + \sqrt{y}) \times (\sqrt{x} + \sqrt{y}) \):

• First: Multiply the first terms of each binomial: \(\sqrt{x} \times \sqrt{x} = x\).
• Outer: Multiply the outer terms: \(\sqrt{x} \times \sqrt{y}\).
• Inner: Multiply the inner terms: \(\sqrt{y} \times \sqrt{x}\), which is also \(\sqrt{xy}\) due to the commutative property.
• Last: Multiply the last terms: \(\sqrt{y} \times \sqrt{y} = y\).

Combine these results, and you get the simplified expression \(x + 2\sqrt{xy} + y\). The FOIL method is a quick and reliable way to ensure all parts of the binomials are multiplied together.

Square Roots

Square roots are mathematical functions that answer the question 'What number, when multiplied by itself, gives the original number?' For example, \(\sqrt{9} = 3\) because \(3 \times 3 = 9\). When dealing with square roots in algebra, you will often encounter variables under the square root symbol.

The simplification of square roots involves two main steps:

• Finding the prime factorization of the number within the root to identify any perfect squares.
• Simplifying expressions by 'taking out' the perfect squares from under the root and leaving the remaining factors inside.

However, with variables, you might need to wait until you have specific numeric values to apply these steps fully. In our problem, \(\sqrt{x}\) and \(\sqrt{y}\) cannot be simplified further without knowing the values of \(x\) and \(y\).

Simplifying Expressions

Simplifying expressions is a key part of solving algebraic equations efficiently. It generally means making the expression as straightforward as possible. This can include combining like terms, factoring, expanding expressions, or reducing fractions to their lowest terms.

In the context of our problem, after using the FOIL method we combine like terms, which in this case are \(\sqrt{xy}\) and \(\sqrt{yx}\). Remember that the order of multiplication does not affect the product due to the commutative property, so these are indeed like terms and are combined to give \(2\sqrt{xy}\). The expression \(x + 2\sqrt{xy} + y\) is the simplified form, as no further reduction or combination of terms is possible.

Commutative Property of Multiplication

The commutative property of multiplication states that the order in which two numbers are multiplied does not change the product. Simply put, \(a \times b = b \times a\). This property is crucial when working with algebraic expressions, as it allows for the rearrangement of terms for easier simplification.

During the solution to our original problem, we used the commutative property to recognize that \(\sqrt{y} \times \sqrt{x}\) is the same as \(\sqrt{x} \times \sqrt{y}\), which simplifies the process of combining like terms after the FOIL method has been applied. Understanding and using this property effectively can make complex algebraic problems much more manageable.