Question

Challenge Problems. Perform the indicated operation and simplify. $$(a+\sqrt{b}) \times(a-\sqrt{b})$$

Short answer

\(a^2 - b\)

Step-by-step solution

Identify the Pattern

Recognize the expression as a difference of squares pattern, which is \(a^2 - b^2\) where our \(a\) is \(a\) and \(b\) is \(\b\rt{b}\).

Apply the Difference of Squares Formula

Use the formula \(a^2 - b^2 = (a + b)(a - b)\) to expand the expression \( (a+\b\rt{b}) \times(a-\b\rt{b}) \) as \( a^2 - (\b\rt{b})^2 \) .

Simplify the Result

Simplify the expression by squaring \(a\) and \(\b\rt{b}\), yielding \( a^2 - b \) since \(\b\rt{b}^2 = b\).

Key concepts

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation. In the exercise with the expression \(a+\sqrt{b}) \times(a-\sqrt{b})\), \(a\) and \(b\) represent variables or constants, and the operation indicated is multiplication. When dealing with algebraic expressions, it's critical to recognize patterns that can be applied to simplify them.

One such pattern is the difference of squares, which is a fundamental identity utilised in simplification. Spotting this pattern is part of algebraic reasoning and allows us to transform complex-looking expressions into more manageable forms. Understanding how to manipulate these expressions requires familiarity with various algebraic identities and the properties of arithmetic operations.

Square Roots

Understanding Square Roots

The square root of a number is a value that, when multiplied by itself, gives the original number. For example, since \(4 \times 4 = 16\), the square root of \(16\) is \(4\), often written as \(\sqrt{16} = 4\). In our exercise, simplifying square roots is essential for the pattern recognition of difference of squares.

The expression \(\sqrt{b}\) may look intimidating at first, but remember that when paired with itself in the form of \((\sqrt{b})^2\), it simplifies back to \(b\). This reversal property of squaring and square roots is pivotal in simplifying expressions and in solving equations involving square roots.

Simplifying Expressions

Breaking Down Simplification

Simplifying expressions is the process of reducing them to their simplest form. This often involves combining like terms, using arithmetic operations, and applying algebraic formulas such as the difference of squares. In the example \((a+\sqrt{b}) \times(a-\sqrt{b})\), by applying the difference of squares formula, simplification is straightforward.

After recognizing the pattern and substituting the values into the formula, we get \(a^2 - (\sqrt{b})^2\), which simplifies to \(a^2 - b\). The key here is to carry out each operation carefully and methodically, ensuring that each step follows logically from the previous one. This final simplified form is much easier to understand and work with for further equations or applications.