What is the simplest form of the expression

Simplified Expression

To simplify the expression $(\sqrt{3} - \sqrt{6})(\sqrt{3} + \sqrt{6})$:

We use the difference of squares formula, which states:

In this case, let:

Applying the formula:

Calculating the squares:

Putting these values back into the equation, we get:

So, the simplest form of the expression is:

Explanation

The difference of squares formula is a fundamental algebraic identity that states:

Concept Breakdown

1. Term Definitions:

   • $a^2$: Square of $a$
   • $b^2$: Square of $b$

2. Difference of Squares: The expression $a^2 - b^2$ represents the difference between the squares of two numbers $a$ and $b$.

3. Factoring: This algebraic identity shows that any difference of squares can be factored into a product of two binomials $(a + b)$ and $(a - b)$.

Example

Let's consider an example to see this formula in action:

Suppose we have $a = 5$ and $b = 3$:

Calculate each part step-by-step:

• First, find the squares:

• Then, the difference:

• Using the formula:

Both methods give the same result, verifying that the difference of squares formula works correctly.

Practical Applications

This formula is extremely useful in various mathematical fields, including:

• Simplifying algebraic expressions
• Solving quadratic equations
• Evaluating integrals in calculus
• Polynomial factorization

Understanding and applying the difference of squares formula provides a vital tool for tackling complex algebraic problems more efficiently.

Explanation of Squaring Radicals

Squaring radicals is a mathematical process that involves raising a square root expression to the power of two. When you square a radical, you are essentially reversing the square root operation.

Basic Concept

If you have a radical expression, like the square root of $x$ (written as $\sqrt{x}$), then squaring this radical results in the original number $x$.

Mathematically, this can be expressed as:

Examples

Let’s consider a few examples to better understand this concept:

1. Squaring the square root of 9:

2. Squaring the square root of 25:

Key Points to Remember

• Cancelling Effect: Squaring a square root "cancels out" the radical, leaving you with the radicand (the number under the square root symbol).

• Non-negative Results: Given that the square root function returns the principal (non-negative) square root, squaring a radical also results in a non-negative number.

Complex Radicals

For cases involving more complex expressions, the same basic principle applies. For instance:

Example: Squaring the square root of $3 + 4$:

Conclusion

Squaring radicals simplifies the expression by removing the square root, essentially "undoing" the square root operation. This fundamental property is widely used in algebra to simplify equations and solve problems involving radicals.