Let"s investigate what happens as soon as negative values appear under the radical symbol (as the radicand) for cube roots and square roots.
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In some cases, negative numbers under a radical symbol are OK. For instance,
Difficulties, however, develop when we look at a problem such as
Yes, (-2) x (-2) x (-2) = -8. No problem.
Nope! (4) x (4) ≠ -16. Nope! (-4) x (-4) ≠ -16.
The square root of a negative number does not exist among the set of Real Numbers.
When difficulties via negatives under a square root initially appeared, mathematicians assumed that a solution did not exist. They observed equations such as x2 + 1 = 0, and also wondered what the solution
|The imaginary number "i" is the square root of negative one.|
An imaginary number possesses the distinct property that when squared, the outcome is negative.
As research with imaginary numbers ongoing, it was found that they actually filled a gap in mathematics and also offered a valuable purpose. Imaginary numbers are vital to the research of scientific researches such as electricity, quantum mechanics, vibration analysis, and also cartography.
When the imaginary i was combined via the collection of Real Numbers, the all encompassing set of Complex Numbers was formed.
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Product Rule wbelow a ≥ 0, b≥ 0
"The square root of a product is equal to the product of the square roots of each aspect."
This theorem permits us to use our method of simplifying radicals.
Imaginary (Unit) Number
Product Rule (extended) where a ≥ 0, b≥ 0 OR a ≥ 0, b 0 but NOT a 0, b 0