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What is the simplest form of the quotient? Simplify by rationalizing the denominator.

√3x¹²y¹⁰ /√5x⁶y³

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Final answer:

The simplest form of the quotient √3x¹²y¹° / √5x⁶y³ after rationalizing the denominator by using the property of Division of Exponentials is (√3/5)x⁶y⁷.

Step-by-step explanation:

The student is asking to simplify the quotient √3x¹²y¹° / √5x⁶y³ by rationalizing the denominator. To do this, we will use the property of Division of Exponentials which tells us to subtract the exponents of like bases when dividing. For instance, x¹² / x⁶ would simplify to x¹²-⁶ = x⁶.

First, we divide the digit term of the numerator by the digit term of the denominator, which gives us √(3/5). Next, we subtract the exponents of the x terms, which gives us x¹²-⁶ = x⁶. Similarly, we subtract the exponents of the y terms, y¹°-³ = y⁷. After putting everything together, our simplified expression becomes (√3/5)x⁶y⁷.

In some cases, further simplification may be possible if there's a need to rationalize the denominator or simplify the square root, but in this case, the square root of a fraction cannot be further simplified.

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