Simplifying radical expression sqrt[75x]/(2 * sqrt[3])

[DustinC]

So heres the question to simplify.

Square root (75x) - top
2 times square root (3) - bottom

The top simplifies into 5 times (square root 3x)
Then the square roots of (3) on both top and bottom cancel each other out.

The problem simplifies to

(5) times square root (x) - top
2 - bottom

Math book turns this into 5/2 then multiplies that to square root (x)

My question is why does it get turned into that instead of

5/2 times square root (x) over 2


The two remains in the denominator throughout the equation simplification. So why did they express the answer without a 2 under the square root of x

 

[JeffM]

So heres the question to simplify.

Square root (75x) - top
2 times square root (3) - bottom

The top simplifies into 5 times (square root 3x)
Then the square roots of (3) on both top and bottom cancel each other out.

The problem simplifies to

(5) times square root (x) - top
2 - bottom

Math book turns this into 5/2 then multiplies that to square root (x)

My question is why does it get turned into that instead of

5/2 times square root (x) over 2 Where did the SECOND denominator of 2 come from?


The two remains in the denominator throughout the equation simplification. So why did they express the answer without a 2 under the square root of x

THE 2 REMAINS UNDER THE 5.

\(\displaystyle \dfrac{\sqrt{75x}}{2\sqrt{3}} = \dfrac{\sqrt{25 * 3} * \sqrt{x}}{2\sqrt{3}} = \dfrac{5 * \sqrt{3} * \sqrt{x}}{2\sqrt{3}} =\)

\(\displaystyle \dfrac{5 * \sqrt{x}}{2} = \dfrac{5 * \sqrt{x}}{2 * 1} = \dfrac{5}{2} * \dfrac{\sqrt{x}}{1} = \frac{5}{2} \sqrt{x}.\)

Got it now?

 

[DustinC]

Hmm

So in the fourth image, that to me looks like the final answer. I honestly don't understand how the denominator of 2 factors into 2x1, and then that you're able to make 2 rational expressions out of it.

The only number in the denominator throught the equation up until the 5th part you put there only has 2 as a denominator. Shouldn't then, any expression in the original numerator also have a denominator of 2 under it?
How are you able to separate the square of (x) and then stick a denominator of 1 under it, when there has been a denominator of 2 under it the whole time.

 

[tkhunny]

It doesn't "factor". 2 * 1 = 2 Simple as that. It matters not how you write it. Some versions of are more convenient for some operations to be more apparent.

 

[JeffM]

So in the fourth image, that to me looks like the final answer. I honestly don't understand how the denominator of 2 factors into 2x1, and then that you're able to make 2 rational expressions out of it.

The only number in the denominator throught the equation up until the 5th part you put there only has 2 as a denominator. Shouldn't then, any expression in the original numerator also have a denominator of 2 under it?
How are you able to separate the square of (x) and then stick a denominator of 1 under it, when there has been a denominator of 2 under it the whole time.

Surely you know that 2 * 1 = 2. In fact, for any number r, 1 * r = r = r * 1.

And you learned about multiplying fractions years ago.

\(\displaystyle \dfrac{wx}{yz} = \dfrac{w}{y} * \dfrac{x}{z} = \dfrac{w}{z} * \dfrac{x}{y}. \)

If what you are asking is whether \(\displaystyle \dfrac{5\sqrt{x}}{2}\) is A correct answer, it is. But a

different answer just as correct is \(\displaystyle \frac{5}{2} \sqrt{x}.\)

Those two expressions mean the same thing. Which is simpler is a matter of opinion. When you are told to simplify an expression, you need to realize that there may be several correct answers, all mathematically equivalent and each simpler than the original.