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Given that a ne b, is it ever possible to have sqrt(a) * sqrt(b) = sqrt(a * b)? Explain.

User TomDestry
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Final answer:

It is possible for sqrt(a) * sqrt(b) to equal sqrt(a * b) when a and b are non-negative, as squaring and square roots are inverse operations. However, if either a or b are negative, this equality doesn't hold due to the nature of square roots producing imaginary numbers.

Step-by-step explanation:

A student asked whether it is ever possible to have sqrt(a) * sqrt(b) = sqrt(a * b), assuming a ≠ b. The answer to this lies in understanding the properties of square roots and exponents. Generally, the square root of a product is equal to the product of the square roots, if a and b are non-negative. This is because according to the properties of exponents, (√a)^{2} = a and (√b)^{2} = b, hence (√a)(√b) = √(ab) when a and b are non-negative. This is because squaring and taking the square root are inverse operations.

However, if a or b are negative, then we must be careful since the square root of a negative number is not a real number but an imaginary one. In such cases, sqrt(a) * sqrt(b) will not necessarily equal sqrt(a * b) due to the properties of complex numbers.

User Anjum Sayed
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