To complete the expression \( ?\sqrt{5} + 3\sqrt{45} = 13\sqrt{?} \), we first simplify \( 3\sqrt{45} \).
We can break down \( \sqrt{45} \): \[ \sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5} = 3\sqrt{5} \]
Thus: \[ 3\sqrt{45} = 3 \times 3\sqrt{5} = 9\sqrt{5} \]
Now we can substitute this back into the original expression: \[ ?sqrt{5} + 9\sqrt{5} = 13\sqrt{?} \]
Combining the terms on the left side gives us: \[ (? + 9)\sqrt{5} = 13\sqrt{?} \]
To find \( ? \), we equate \( ? + 9 \) to \( 13 \) (since the coefficients of \( \sqrt{5} \) and \( \sqrt{?} \) should match), meaning there’s a relation between the terms: \[ ? + 9 = 13 \]
Solving for \( ? \): \[ ? = 13 - 9 = 4 \]
Now we can rewrite \( 13\sqrt{?} \): \[ 13\sqrt{?} = 13\sqrt{4} = 13 \cdot 2 = 26 \]
Thus, the completed expression is: \[ 4\sqrt{5} + 9\sqrt{5} = 26 \]
So the final answer is: \[ \boxed{4} \]
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