How is the expression (a + √b)(a - √b) simplified?

How is the expression (a + √b)(a - √b) simplified?

• A. (a² - b)
• B. (a² + b)
• C. (a² - 2a√b + b)
• D. (a² - a√b)

Answer: (A)

The expression \((a + \sqrt{b})(a - \sqrt{b})\) can be simplified using the difference of squares formula, which states that \((x + y)(x - y) = x^2 - y^2\). Here, \(x\) is \(a\) and \(y\) is \(\sqrt{b}\). When applying this formula, we substitute \(x\) and \(y\) into the formula: 1. Calculate \(x^2\): - This is \(a^2\) since \(x = a\). 2. Calculate \(y^2\): - This is \((\sqrt{b})^2\), which simplifies to \(b\). After performing the calculations, the expression simplifies to: \[ a^2 - b \] This matches the first choice provided. The other choices do not represent the correct application of the difference of squares. For example, \(a^2 + b\) adds the squares instead of subtracting them, while the others introduce unnecessary terms or incorrect combinations of \(a\) and \(b\). Therefore, \((a + \sqrt{b})(a

The expression ((a + \sqrt{b})(a - \sqrt{b})) can be simplified using the difference of squares formula, which states that ((x + y)(x - y) = x^2 - y^2). Here, (x) is (a) and (y) is (\sqrt{b}).

When applying this formula, we substitute (x) and (y) into the formula:

1. Calculate (x^2):

• This is (a^2) since (x = a).

2. Calculate (y^2):

• This is ((\sqrt{b})^2), which simplifies to (b).

After performing the calculations, the expression simplifies to:

[

a^2 - b

]

This matches the first choice provided. The other choices do not represent the correct application of the difference of squares. For example, (a^2 + b) adds the squares instead of subtracting them, while the others introduce unnecessary terms or incorrect combinations of (a) and (b). Therefore, ((a + \sqrt{b})(a