To solve the expression \( ?\sqrt{5} + 3\sqrt{45} = 13\sqrt{?} \), we will simplify the left-hand side and analyze the right-hand side.
First, let's simplify \( 3\sqrt{45} \): \[ \sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \cdot \sqrt{5} = 3\sqrt{5} \] Thus, \[ 3\sqrt{45} = 3 \times 3\sqrt{5} = 9\sqrt{5} \]
Now we substitute this back into the expression: \[ ?\sqrt{5} + 9\sqrt{5} = 13\sqrt{?} \]
Now, we can combine like terms on the left side: \[ (? + 9)\sqrt{5} = 13\sqrt{?} \]
For the equality to hold, we need both sides to be expressed with the same radical. Thus, we must have: \[ ? + 9 = 13 \quad \text{(the coefficients of }\sqrt{5}\text{ and }\sqrt{?}\text{ must be the same)} \] Solving for \( ? \): \[ ? = 13 - 9 = 4 \]
Now we substitute back to find \( ? \) in \( 13\sqrt{?} \): Since we found \( ? = 4 \): \[ 13\sqrt{?} = 13\sqrt{4} = 13 \times 2 = 26 \]
This tells us that \( 13\sqrt{?} = 26 \) confirms that \( \sqrt{?} = 2 \), leading to \( ? = 4 \).
As a summary, we can rewrite our solution: \[ ? = 4 \quad \text{and we need to find another integer for the right side, which corresponds to }\sqrt{5}\text{.} \]
Thus, our final values for the question marks are: \[ ? = 5 \quad \text{(from }\sqrt{5} \text{) and } ? = 4 \text{ (duplicate for }\sqrt{4}\text{)} \]
To conclude:
- The first question mark corresponds to the coefficient of \( \sqrt{5} \), which is \( 4 \).
- The second question mark corresponds to \( ? \) in \( 13\sqrt{?} \), which evaluates to \( 4 \).
Therefore, the final answers are: \[ \text{Answers: } 4 \text{ and } 5. \]
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