To solve for the values of c and d that satisfy the equation 162x^c y^5 = 3x^2 y^{6d}, we can start by comparing the coefficients, the powers of x, and the powers of y on both sides of the equation.
Step 1: Compare the coefficients
The coefficient on the left side is 162 and on the right side is 3. To find the relationship between them, we divide both sides:
162 / 3 = 54
Thus, the coefficients suggest that:
162 = 54 * 3
Step 2: Compare the powers of x
On the left side, we have x^c and on the right side, we have x^2. For the equation to hold true, the exponents must be equal:
c = 2
Step 3: Compare the powers of y
Next, we look at the powers of y. On the left side, we have y^5 and on the right side, we have y^{6d}. Again, we equate the exponents:
5 = 6d
Now, solve for d:
d = 5/6
Final Values:
Summarizing the findings:
- c = 2
- d = 5/6
Thus, the values of c and d that make the equation 162x^c y^5 = 3x^2 y^{6d} true are c = 2 and d = 5/6.