Question

Let’s explore a quotient raised to an exponent

Answer 1

The answer is 216/y^3

Answer 2

You can think of exponents as abbreviations for repeated multiplication. By that definition:

$\big(\frac{6}{y}\big) ^3 =\frac{6}{y} \times\frac{6}{y} \times\frac{6}{y}$

Which, by the properties of fraction multiplication, is the same thing as

$\frac{6\times6\times6}{y\times y\times y}$

whiiiich, going back to our abbreviations, is the same thing as

$\frac{6^3}{y^3}$

And since 6³ = 216, our final simplified answer is

$\frac{216}{y^3}$

The most important discovery in this problem is the property that

$\big( \frac{x}{y}\big)^n=\frac{x^n}{y^n}$

You could almost say that the exponent gets "distributed" to the numerator and denominator, and in fact, any exponents will have this property of "distributing" across multiplication or division.