SOLUTION: Evaluate the three radicals. Though they look similar, each one is different. {{{sqrt(16/49)}}} - {{{sqrt(16/49)}}} {{{sqrt(-16/49)}}} -{{{sqrt(-16/49)}}} Which of the abo

Algebra ->  Square-cubic-other-roots -> SOLUTION: Evaluate the three radicals. Though they look similar, each one is different. {{{sqrt(16/49)}}} - {{{sqrt(16/49)}}} {{{sqrt(-16/49)}}} -{{{sqrt(-16/49)}}} Which of the abo      Log On


   

Question 1202968: Evaluate the three radicals. Though they look similar, each one is different.
sqrt%2816%2F49%29
- sqrt%2816%2F49%29
sqrt%28-16%2F49%29
-sqrt%28-16%2F49%29
Which of the above square roots can be simplified with a real number answer and what is the simplified value?
Which of the above square roots cannot be simplified and why can they not be simplified?
Answer by math_tutor2020(3816) About Me  (Show Source):
You can put this solution on YOUR website!

sqrt%2816%2F49%29+=+sqrt%28%284%5E2%29%2F%287%5E2%29%29

sqrt%2816%2F49%29+=+%28sqrt%284%5E2%29%29%2F%28sqrt%287%5E2%29%29

sqrt%2816%2F49%29+=+4%2F7 The square root and squaring exponents cancel out

Through similar steps,
-sqrt%2816%2F49%29+=+-4%2F7

Therefore, the first two items can be simplified to a real number result.

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The last two items cannot be simplified to a real number result because of the negative under the square root.


But if we involve the imaginary number i+=+sqrt%28-1%29, then we can say
sqrt%28-16%2F49%29+=+%284%2F7%29i
and
-sqrt%28-16%2F49%29+=+-%284%2F7%29i
Both of these results are complex numbers of the form a+bi

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The reason a negative number is not allowed under the square root is to note how x%5E2 produces nonnegative values when x is a real number.

Examples:
x+=+3 leads to x%5E2+=+3%5E2+=+3%2A3+=+9 which is nonnegative
x+=+0 leads to x%5E2+=+0%5E2+=+0%2A0+=+0 which is nonnegative
x+=+-5 leads to x%5E2+=+%28-5%29%5E2+=+%28-5%29%2A%28-5%29+=+25 which is nonnegative
There is no way to have x%5E2 be negative when x is real.

So if x%5E2+=+y then x+=+sqrt%28y%29 shows that we can only plug in nonnegative values for y.