Which of the following represents `(root6(x^(10)y ^(12)))/(root 3 (x ^(5)y ^(6)))` written in simplest from, given that `x gt 0 and y gt 0` ?

Options

A. 1
B. 2
C. `x ^(2) y ^(3) sqrtx`
D. `xy ^(2)root 3(x ^(2))`

Answer

To simplify the expression \(\frac{\sqrt[6]{x^{10}y^{12}}}{\sqrt[3]{x^{5}y^{6}}}\), we will follow these steps:

### Step 1: Rewrite the roots using exponent notation
We can express the roots in terms of exponents. The sixth root can be written as raising to the power of \(\frac{1}{6}\) and the cube root can be written as raising to the power of \(\frac{1}{3}\).

\[ \frac{\sqrt[6]{x^{10}y^{12}}}{\sqrt[3]{x^{5}y^{6}}} = \frac{(x^{10}y^{12})^{\frac{1}{6}}}{(x^{5}y^{6})^{\frac{1}{3}}} \]

### Step 2: Apply the power of a product property
Using the property \((ab)^n = a^n b^n\), we can distribute the exponent in both the numerator and the denominator.

\[ = \frac{x^{10 \cdot \frac{1}{6}} y^{12 \cdot \frac{1}{6}}}{x^{5 \cdot \frac{1}{3}} y^{6 \cdot \frac{1}{3}}} \]

### Step 3: Simplify the exponents
Now, we simplify the exponents in the numerator and the denominator.

- In the numerator:
- \(x^{10 \cdot \frac{1}{6}} = x^{\frac{10}{6}} = x^{\frac{5}{3}}\)
- \(y^{12 \cdot \frac{1}{6}} = y^{\frac{12}{6}} = y^{2}\)

- In the denominator:
- \(x^{5 \cdot \frac{1}{3}} = x^{\frac{5}{3}}\)
- \(y^{6 \cdot \frac{1}{3}} = y^{2}\)

So, we have:

\[ = \frac{x^{\frac{5}{3}} y^{2}}{x^{\frac{5}{3}} y^{2}} \]

### Step 4: Cancel out the common terms
Since the numerator and denominator are the same, they cancel each other out.

\[ = 1 \]

### Final Answer:
Thus, the expression simplifies to \(1\).