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.