What values of c and d make the equation true: 162x^c y^5 = 3x^2 y^6 y^d?

To determine the values of c and d that make the equation

162x^cy⁵ = 3x²y⁶y^d,

we first need to simplify the right-hand side. The term y⁶y^d can be combined using the properties of exponents:

y⁶y^d = y^{6 + d}.

Now, the equation looks like this:

162x^cy⁵ = 3x²y^{6 + d}.

Next, we compare the coefficients on both sides. The coefficient on the left is 162 and on the right is 3:

162 = 3.

To find the relationship, we can divide both sides by 3:

162 / 3 = 54.

This shows that the coefficient comparison is not directly equal, but we can consider the values and the factor:

162 = 3 * 54.

Now, we analyze the exponents for x. On the left, we have c, and on the right, we have 2. Thus, we set:

c = 2.

Now let’s check the y variables. The left side has 5, while the right side has 6 + d. We equate these:

5 = 6 + d.

Rearranging this gives:

d = 5 – 6 = -1.

Thus, the values that make the equation true are:

c = 2 and d = -1.