To solve the equation √648 = √(2a + 3b), we first need to simplify the left side.
We can express 648 as a product of its prime factors:
648 = 2^3 * 3^4
This means:
√648 = √(2^3 * 3^4) = √(2^2 * 3^4 * 2) = 2 * 3^2 * √2 = 18√2.
So, to make our equation true:
√(2a + 3b) must equal 18√2.
Squaring both sides gives:
2a + 3b = (18√2)² = 324 * 2 = 648.
Now we are looking for values of a and b that satisfy:
2a + 3b = 648.
There are infinitely many combinations of (a, b) that can satisfy this equation. For example:
- If a = 0, then 3b = 648 → b = 216.
- If a = 324, then 3b = 0 → b = 0.
In conclusion, the values of a and b can vary as long as they satisfy the equation 2a + 3b = 648.