I am very confused on how to rationalize a denominator, it just gets harder and harder.will be using the V symbol again as a square root symbol. I get how to rationalize something simple as V2/V7, I would get V14/7 as the answer, but I get confused on how to solve 1/V6 +2, this is the way I did it, and I know it's wrong.

Question

I am very confused on how to rationalize a denominator, it just gets harder and harder.will be using the V symbol again as a square root symbol. I get how to rationalize something simple as V2/V7, I would get V14/7 as the answer, but I get confused on how to solve 1/V6 +2, this is the way I did it, and I know it's wrong.

1/V6 + 2 * V6 - 2/V6 - 2
1V6-2/V36 - 4
1V6 - 2/6-4
1V6 - 2/2

I don't get how to solve from here, would the 2's cancel out? and I'm left with 1V6.

And this is an even harder one...
V30 + 3V3/V30 - 3V3

Answer 1

(a+b)(a-b) = a² - b²

(√6 + 2)(√6 - 2) = (√6)² - 2² = 6-4 = 2

So, you end up with (√6 - 2)/2 = √6/2 - 1

Your lack of parentheses got you the wrong answer.

(√30 + 3√3)/((√30 - 3√3) * (√30 + 3√3)/(√30 + 3√3)

= (30 + 6√30 + 27) / (30-27)
= (57 + 6√30)/3 = 19 + 2√30

Answer 2

Rationalizing the denominator can indeed be confusing at times, but with a bit of understanding, you'll be able to tackle more complex examples as well. Let's go through your questions step by step:

1. Rationalizing 1/V6 + 2:
To rationalize this expression, we need to eliminate the denominator that contains the square root (V6). The approach you used is almost correct, but you made a small mistake in the simplification step.

Here's the correct step-by-step process:

1/V6 + 2 * (V6 - 2)/(V6 - 2)
= (V6 - 2 + 2)/(V6 - 2)
= V6/V6
= 1

So the final simplified expression is 1.

You were on the right track, but instead of dividing 2 by 6-4, you simply had to subtract 2 from V6. Remember that when multiplying fractions, you should multiply the numerators and denominators separately.

2. Rationalizing V30 + 3V3/V30 - 3V3:
In this example, we need to eliminate the denominator that contains both V30 and V3.

Here's how you can approach it:

V30 + 3V3 * (V30 + 3V3)/(V30 + 3V3)
= (V30 * (V30 + 3V3) + 3V3 * (V30 + 3V3))/(V30 - (3V3)^2)
= (V30^2 + V30 * 3V3 + V30 * 3V3 + 3V3 * 3V3)/(V30 - 3^2 * V3^2)
= (30 + 6V10 + 6V10 + 9)/(30 - 9 * 3)
= (39 + 12V10)/(30 - 27)
= (39 + 12V10)/3
= (13 + 4V10)

So the final simplified expression is 13 + 4V10.

I hope this clarifies the steps you need to follow to rationalize denominators. Keep practicing, and it will become easier! Let me know if you have any more questions.