### Video Transcript

Simplify the expression two π₯ to the power of negative one-quarter over three π¦ to the power of two-thirds all squared, where π₯ and π¦ are positive numbers.

We will answer this question using our knowledge of the laws of exponents or indices. We begin by recalling that when we raise a fraction to a power, we can simply raise the numerator and denominator to that power separately. π over π to the πth power is equal to π to the πth power over π to the πth power. This means that we can rewrite our expression as two π₯ to the power of negative one-quarter squared divided by three π¦ to the power of two-thirds squared.

At this stage, we might be tempted to use the power rule of exponents, which states that π to the power of π raised to the power of π is equal to π to the power of π multiplied by π. However, since weβre squaring the numerator and denominator, we will simply multiply two π₯ to the power of negative one-quarter by itself and also multiply three π¦ to the power of two-thirds by itself.

We can now use the rule that states π to the power of π multiplied by π to the power of π is equal to π to the power of π plus π. On the numerator, we need to add negative one-quarter and negative one-quarter. This is equal to negative two-quarters which simplifies to negative one-half. Two π₯ to the power of negative one-quarter multiplied by two π₯ to the power of negative one-quarter is therefore equal to four π₯ to the power of negative one-half. We can repeat this process on the denominator, giving us a power or exponent of four-thirds. Three π¦ to the power of two-thirds squared is equal to nine π¦ to the power of four-thirds.

At this stage, we might think we have finished. However, we can also write the term with a negative exponent in a simplified form. π to the power of negative π is equal to one over π to the power of π. This means that π₯ to the power of negative one-half is equal to one over π₯ to the power of one-half. And we can therefore rewrite our expression as four over nine π₯ to the power of a half π¦ to the power of four-thirds. This is the simplified version of two π₯ to the power of negative one-quarter over three π¦ to the power of two-thirds all squared.