Question

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

Short answer

\(x\sqrt{y} + y\)

Step-by-step solution

Apply the Distributive Property

Use the distributive property to multiply each term inside the parentheses by \(\sqrt{y}\).

Multiply the Terms

Multiply \(x\) by \(\sqrt{y}\) to get \(x\sqrt{y}\) and multiply \(\sqrt{y}\) by \(\sqrt{y}\) to get \(y\) since the square root of a number times itself gives the number.

Combine the Results

Combine the products of the multiplication to write the simplified expression.

Key concepts

Distributive Property

The distributive property is a cornerstone in algebra, allowing us to simplify expressions involving the multiplication of a sum or difference. Specifically, it states that multiplying a sum by a number gives the same result as multiplying each addend by the number and then summing the products. In formal terms, this property is expressed as: \[ a(b + c) = ab + ac. \]

When you come across an expression like \( (x + \sqrt{y}) \times \sqrt{y} \) as in our challenge problem, you'll want to apply this property. It means that \( x \times \sqrt{y} \) and \( \sqrt{y} \times \sqrt{y} \) should be calculated independently before adding them together. This way, you break down a potentially complicated problem into easier, more manageable parts.

Square Root Multiplication

Multiplying square roots may seem daunting at first, but it follows a simple rule: the product of two square roots is the square root of the product of the numbers. So, \( \sqrt{a} \times \sqrt{b} = \sqrt{ab} \). Additionally, when you multiply a square root by itself, such as \( \sqrt{y} \times \sqrt{y} \), you're essentially calculating the square root of \( y \) squared, which simplifies back to \( y \).

Important note:

Square root multiplication is only straightforward when dealing with non-negative numbers, as the square root of a negative number introduces the concept of imaginary numbers, which is beyond the scope of our current discussion.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations that represent a specific mathematical quantity. When working with algebraic expressions, it is crucial to understand the order of operations and how to simplify expressions through techniques such as combining like terms and factoring.

Simplifying Expressions:

In the context of our example, simplifying the expression \((x + \sqrt{y}) \times \sqrt{y}\) can be thought of as 'cleaning up' the expression to its most basic form without changing its value. This often makes it easier to evaluate or manipulate in further calculations. Remember, practice is key to becoming proficient with algebraic expressions, so don't be discouraged by initially complex-looking problems!