Operations with radical expressions:

Simplify each quotient

7/sq rt 2 - sq rt 3

Please solve and explain how you did it. What do you do when the denominator is a negative?

7/(√2-√3)

since (a-b)(a+b) = a^2-b^2,

= 7/(√2-√3) * (√2+√3)/(√2+√3)
= 7(√2+√3)/(2-3)
= -7(√2+√3)

To simplify the quotient 7/(√2 - √3), we need to rationalize the denominator. We achieve this by multiplying the numerator and denominator by the conjugate of the denominator, which is (√2 + √3).

Let's go through the steps:

Step 1: Multiply the numerator and denominator by the conjugate of the denominator (√2 + √3):

(7/(√2 - √3)) * (√2 + √3)/(√2 + √3)

Step 2: Simplify the numerator by using the distributive property:

7(√2 + √3)

Step 3: Simplify the denominator using the difference of squares formula:

(√2 - √3)(√2 + √3) = (√2)^2 - (√3)^2 = 2 - 3 = -1

So, we end up with:

7(√2 + √3)/-1

Step 4: Simplify further by dividing numerator and denominator by -1:

-7(√2 + √3)

Therefore, the simplified quotient 7/(√2 - √3) is -7(√2 + √3).

Now, regarding your question about a negative denominator, we can still follow the same steps to simplify the expression. The only difference is that when we multiply the numerator and denominator by the conjugate, we have a negative sign in front of the denominator after simplification.

If the denominator is a negative, it does not change the process of simplifying the expression, but it does change the sign of the simplified expression.

To simplify the quotient 7/(√2 - √3), we need to rationalize the denominator. Rationalizing the denominator means removing any radical expression from the denominator.

To do this, we need to multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of √2 - √3 is √2 + √3.

Multiplying the numerator and denominator by the conjugate, we get:
7 * (√2 + √3) / ((√2 - √3) * (√2 + √3))

Now, let's simplify this expression step by step:

In the denominator, we can apply the difference of squares formula:
(√2 - √3) * (√2 + √3) = (√2)^2 - (√3)^2
= 2 - 3
= -1

So, the denominator becomes -1.

In the numerator, we can distribute the 7 to both terms:
7 * (√2 + √3) = 7√2 + 7√3

Now we have:
(7√2 + 7√3) / -1

Since the denominator is -1, we can simply write -1 as the negative sign in front of the fraction. Therefore, the expression simplifies to:
- (7√2 + 7√3)

To summarize:
7/(√2 - √3) simplifies to - (7√2 + 7√3).

Regarding your question about when the denominator is negative, you can see that we simply wrote the negative sign in front of the fraction. In general, when the denominator is negative, you can directly write the negative sign in front of the fraction without having to change any aspect of the expression.