Math Olympiad Question – Solve for 32^x · 8^y

Question:

Find the value of 32^x · 8^y if 5x + 3y = 3

 Step-by-Step Solution

Step-by-Step Solution:

1. Given equation:

   $$5x + 3y = 3$$

2. Write bases in powers of 2:

   $$32 = 2^5, \quad 8 = 2^3$$

3. Substitute into the expression:

   $$32^x \cdot 8^y = (2^5)^x \cdot (2^3)^y$$

4. Simplify powers:

   $$= 2^{5x} \cdot 2^{3y}$$

5. Combine exponents (since base is same):

   $$= 2^{5x + 3y}$$

6. From the given equation:

   $$5x + 3y = 3$$

7. Substitute:

   $$2^{5x+3y} = 2^3$$

8. Simplify:

   $$=8$$

✅ Final Answer

                   32^x · 8^y = 8