 Negative Value Underthe Square Root Radical ubraintv-jp.com Topical Overview | Algebra 1 Summary | MathBits" Teacher Reresources Terms of Use Contact Person: Donna Roberts 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, is not a trouble given that (-2) • (-2) • (-2) = -8, making the answer -2. In cube root troubles, it is feasible to multiply an unfavorable value times itself three times and also gain a negative answer.

Difficulties, however, develop when we look at a problem such as . This square root difficulty is asking for a number multiplied times itself that will certainly offer a product (answer) of -16. There simply is no method to multiply a number times itself and gain an unfavorable outcome. Consider: (4) • (4) = 16 and also (-4) • (-4) = 16.

CUBE ROOTS: BUT
SQUARE ROOTS: Yes, (-2) x (-2) x (-2) = -8. No problem.

Nope! (4) x (4) ≠ -16. Nope! (-4) x (-4) ≠ -16. Square roots are the culprits! The challenges arise when you encounter an adverse worth under a square root. It is not possible to square a worth (multiply it times itself) and also arrive at a negative value. So, what carry out we do? 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 really supposed. In an initiative to deal with this trouble, mathematicians "created" a brand-new number, i, which was described as an "imaginary number", since it was not in the collection of "Real Numbers". This brand-new number was viewed with much skepticism. The imaginary number initially appeared in print in the year 1545. 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. Consider: The process of simplifying a radical containing an adverse variable is the very same as normal radical simplification. The only distinction is that the will certainly be reinserted via an "i ".

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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"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 