Problem 62 - Algebra A Combined Function

Question

Multiply. Then simplify if possible. Assume that all variables represent positive real numbers. $$ (\sqrt{x}-y)(\sqrt{x}+y) $$

Step-by-Step Solution

Answer

The simplified expression is \(x - y^2\).

Step 1: Identify the Expression Type

The given expression is \[(\sqrt{x} - y)(\sqrt{x} + y)\]This expression is in the form \[(a-b)(a+b)\]which is a difference of squares.

Step 2: Apply the Difference of Squares Formula

For any expression of the form \[(a-b)(a+b)\]we use the formula\[a^2 - b^2\].Here, in our expression, \(a = \sqrt{x}\) and \(b = y\). So the expression becomes \[(\sqrt{x})^2 - y^2\].

Step 3: Simplify the Expression

Now, simplify \[(\sqrt{x})^2 - y^2\].Since \[(\sqrt{x})^2 = x\], the expression simplifies to \[x - y^2\].Thus, \[(\sqrt{x} - y)(\sqrt{x} + y) = x - y^2\].

Key Concepts

Multiplication of Expressions

When we multiply expressions, we apply distribution or a formula to combine terms. In the exercise, the expression \((\sqrt{x} - y)(\sqrt{x} + y)\) is multiplied using a special method known as the "difference of squares". This is a simple way to multiply expressions that follow the pattern \((a - b)(a + b)\). The formula for difference of squares is:

• \((a - b)(a + b) = a^2 - b^2\)

You use this formula when you see the pattern of two similar terms, just with opposite signs in the middle. So in our case, \(a = \sqrt{x}\) and \(b = y\), so the formula becomes \((\sqrt{x})^2 - y^2\). This quickly shows how multiplication reduces using this tool.

Simplifying Expressions

After multiplying expressions, simplification helps us make them easier to read and understand. Simplifying involves combining like terms or reducing the expression to its simplest form. In the example provided, once we apply the difference of squares formula, we get \((\sqrt{x})^2 - y^2\). Now, let's break this down:

• \((\sqrt{x})^2\) simplifies to \(x\) because the square root and the square cancel each other out.

Thus, the whole expression simplifies to \(x - y^2\). It’s essential to check if further simplification is possible, but in this case, \(x - y^2\) is already in its simplest form as no like terms can be combined.

Square Roots

Square roots appear often in algebra and are essential in simplification and solving expressions. A square root is a value that, when multiplied by itself, gives the original number. For instance, \(\sqrt{x}\) is a number which when squared results in \(x\). Important points about square roots:

• Squaring a square root, like \((\sqrt{x})^2\), results in the original value: \(x\).
• Square roots are often involved in formulas, like the difference of squares.

In the exercise, understanding square roots is key to simplifying the expression to \(x\) from \((\sqrt{x})^2\). Knowing this makes multiplying and simplifying expressions more approachable.