Solving A Tricky Radical Expression: A Step-by-Step Guide

Hey guys! Today, we're diving into a fun math problem involving radicals. This one might look a bit intimidating at first, but don't worry, we'll break it down step by step and make it super easy to understand. We're going to tackle the expression: [(2/12^(1/2)) + (5/75^(1/2)) - (4/192^(1/2)) + (12/108^(1/2))] * (12/6^(1/2)) and figure out which of the given options is the correct answer. The options are: A. -2^(1/2) B. 2^(1/2) C. 3^(1/2) D. 6^(1/2).

Breaking Down the Expression

First things first, let's simplify those radicals in the denominators. Remember, a radical like 12^(1/2) is the same as the square root of 12, which we can write as √12. So, let's rewrite the expression using square root notation to make things a bit clearer:

[(2/√12) + (5/√75) - (4/√192) + (12/√108)] * (12/√6)

Now, we need to simplify each of the square roots by finding the largest perfect square that divides each number. This will help us to pull out factors from under the radical sign. It's like giving the numbers a little radical makeover, haha!

Simplifying the Radicals

• √12: The largest perfect square that divides 12 is 4 (since 4 * 3 = 12). So, √12 = √(4 * 3) = √4 * √3 = 2√3.
• √75: The largest perfect square that divides 75 is 25 (since 25 * 3 = 75). So, √75 = √(25 * 3) = √25 * √3 = 5√3.
• √192: The largest perfect square that divides 192 is 64 (since 64 * 3 = 192). So, √192 = √(64 * 3) = √64 * √3 = 8√3.
• √108: The largest perfect square that divides 108 is 36 (since 36 * 3 = 108). So, √108 = √(36 * 3) = √36 * √3 = 6√3.
• √6: This one is already in its simplest form because 6 doesn't have any perfect square factors other than 1. So, √6 stays as √6.

Now, let's substitute these simplified radicals back into our expression:

[(2/2√3) + (5/5√3) - (4/8√3) + (12/6√3)] * (12/√6)

Simplifying the Fractions Inside the Brackets

Okay, things are looking a little better now! We have simplified radicals in all the denominators. Let's simplify the fractions inside the brackets. This means reducing each fraction to its lowest terms. Think of it as cleaning up the fraction clutter!

• (2/2√3) simplifies to 1/√3 (divide both numerator and denominator by 2).
• (5/5√3) simplifies to 1/√3 (divide both numerator and denominator by 5).
• (4/8√3) simplifies to 1/2√3 (divide both numerator and denominator by 4).
• (12/6√3) simplifies to 2/√3 (divide both numerator and denominator by 6).

Our expression now looks like this:

[(1/√3) + (1/√3) - (1/2√3) + (2/√3)] * (12/√6)

Combining the Fractions Inside the Brackets

Now, we need to add and subtract the fractions inside the brackets. To do this, we need a common denominator. Notice that all the fractions have √3 in the denominator, but one of them has a 2 as well (2√3). So, the least common denominator (LCD) for these fractions is 2√3.

Let's rewrite each fraction with the common denominator of 2√3:

• (1/√3) becomes (2/2√3) (multiply numerator and denominator by 2).
• (1/√3) becomes (2/2√3) (multiply numerator and denominator by 2).
• (1/2√3) stays as (1/2√3).
• (2/√3) becomes (4/2√3) (multiply numerator and denominator by 2).

Our expression inside the brackets now looks like this:

(2/2√3) + (2/2√3) - (1/2√3) + (4/2√3)

Now we can combine the numerators, keeping the common denominator:

(2 + 2 - 1 + 4) / 2√3 = 7/2√3

So, our expression now simplifies to:

(7/2√3) * (12/√6)

Multiplying the Fractions

Now, we're down to multiplying two fractions! This is relatively straightforward. We simply multiply the numerators together and the denominators together:

(7 * 12) / (2√3 * √6) = 84 / (2 * √(3 * 6)) = 84 / (2√18)

Simplifying the Resulting Radical

We have a new radical to simplify! Let's simplify √18. The largest perfect square that divides 18 is 9 (since 9 * 2 = 18). So, √18 = √(9 * 2) = √9 * √2 = 3√2.

Substituting this back into our expression, we get:

84 / (2 * 3√2) = 84 / 6√2

Rationalizing the Denominator

We're almost there! But, mathematicians prefer to not have radicals in the denominator of a fraction. This process is called rationalizing the denominator. To do this, we multiply both the numerator and the denominator by the radical in the denominator (which is √2 in this case):

(84 / 6√2) * (√2 / √2) = (84√2) / (6 * 2) = (84√2) / 12

Final Simplification

Finally, we can simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 12:

(84√2) / 12 = 7√2

Oops! It seems like we made a small error somewhere. 7√2 isn't one of the options. Let's go back and carefully review our steps, especially the multiplication and simplification of the fractions. It's super common to make a tiny mistake in these long calculations, so don't get discouraged!

Alright, after carefully reviewing, I spotted a small mistake! When we simplified 84 / 6√2, we should have divided 84 by 6 first, which gives us 14. So, we have 14/√2. Now, let's rationalize the denominator:

(14/√2) * (√2/√2) = 14√2 / 2 = 7√2

Still not matching the options! Let's rewind a bit further... Ah, I see it! When we simplified the initial expression, we ended up with (7/2√3) * (12/√6). Let's rewrite this and be extra careful with our simplification:

(7/2√3) * (12/√6) = (7 * 12) / (2√3 * √6) = 84 / (2√18)

Now, √18 = √(9 * 2) = 3√2. So we have:

84 / (2 * 3√2) = 84 / 6√2

Dividing 84 by 6 gives us 14, so we have:

14/√2

Now, rationalize the denominator by multiplying both the numerator and denominator by √2:

(14/√2) * (√2/√2) = 14√2 / 2 = 7√2

Still not in the options! Okay, let's try simplifying 14/√2 directly by multiplying top and bottom by √2:

(14/√2) * (√2/√2) = 14√2 / 2 = 7√2

We keep getting 7√2, which isn't one of the options. This suggests there might be a clever simplification we're missing, or perhaps we need to manipulate the answer to match one of the options. Let's look at the options again:

A. -√2 B. √2 C. √3 D. √6

Our result is 7√2. Let’s think… We haven’t made a mistake in the calculations, so maybe the answer is hiding in plain sight. Did we miss a negative sign somewhere? Or can we manipulate 7√2 to look like one of the options?

Sometimes in math problems, you have to massage the answer a bit to match the given choices. Let's think about how 7√2 relates to the options. It seems we've hit a snag! Let’s backtrack and meticulously check each step one more time. This is a great reminder that even small errors can lead to big discrepancies, haha!

Okay, after a super thorough review (and maybe a little bit of staring intensely at the calculations!), the mistake was found! It's a classic one: a sign error. When combining the fractions inside the brackets, we had: (2/2√3) + (2/2√3) - (1/2√3) + (4/2√3). We correctly calculated the numerator as 2 + 2 - 1 + 4 = 7. But, let's go back to the original expression before we found the common denominator:

[(1/√3) + (1/√3) - (1/2√3) + (2/√3)]

To get the common denominator of 2√3, we had:

(2/2√3) + (2/2√3) - (1/2√3) + (4/2√3)

Everything looks correct up to this point. The sum (2 + 2 - 1 + 4) is indeed 7. So, we have 7/2√3. Then, we multiply by 12/√6:

(7/2√3) * (12/√6) = 84 / 2√18 = 84 / (2 * 3√2) = 84 / 6√2 = 14/√2

Rationalizing the denominator, we get:

(14/√2) * (√2/√2) = 14√2 / 2 = 7√2

Okay, we are still at 7√2. The options are A. -√2, B. √2, C. √3, D. √6. It seems there might be an issue with the options provided or a misinterpretation of the problem. Given our calculations, none of the provided options match the correct answer, which is 7√2.

Conclusion

So, guys, after all that careful calculation and double-checking, it looks like the correct answer, 7√2, isn't actually one of the options given. This happens sometimes in math, and it's a good reminder to always trust your work! It's possible there was a typo in the options, or maybe the question was designed to trick us. But, by breaking down the problem step by step and simplifying each radical and fraction carefully, we arrived at the correct answer. Keep practicing, and you'll become a radical-solving pro in no time! Hahaha! Remember, math is all about the journey, not just the destination. And sometimes, the journey leads us to discover that the map was wrong all along!