Raising a Product or a Quotient to a Power.

(xy4)9

= x^9 y^36

To raise a product or a quotient to a power, you need to apply the power to each term or factor individually.

In the given expression (xy^4)^9, we have a product of two terms: xy^4. To raise this product to the power of 9, we will apply the power to each term separately.

First, let's focus on the first term, x. To raise x to the power of 9, we simply multiply x by itself 9 times:

x^9 = x * x * x * x * x * x * x * x * x

Now, let's look at the second term, y^4. To raise y^4 to the power of 9, we also multiply y^4 by itself 9 times:

(y^4)^9 = (y^4) * (y^4) * (y^4) * (y^4) * (y^4) * (y^4) * (y^4) * (y^4) * (y^4)

When multiplying powers with the same base, you add the exponents, so in this case, we add the exponents of y, which is 4:

(y^4)^9 = y^(4 * 9) = y^36

Finally, we combine the results for x and y^36:

(xy^4)^9 = x^9 * (y^4)^9 = x^9 * y^36

So, the expression (xy^4)^9 simplifies to x^9 * y^36.