To simplify the expression \(10\sqrt{5} + 3\sqrt{45}\) and express it in the form \(13\sqrt{x}\), we first simplify \(3\sqrt{45}\).
First, we can simplify \(\sqrt{45}\):
\[ \sqrt{45} = \sqrt{9 \cdot 5} = \sqrt{9} \cdot \sqrt{5} = 3\sqrt{5} \]
Now we can substitute this back into the expression:
\[ 3\sqrt{45} = 3(3\sqrt{5}) = 9\sqrt{5} \]
Now we can add this to \(10\sqrt{5}\):
\[ 10\sqrt{5} + 9\sqrt{5} = (10 + 9)\sqrt{5} = 19\sqrt{5} \]
To have it in the form \(13\sqrt{x}\), we can express \(19\sqrt{5}\) as follows:
\[ 19\sqrt{5} = 13\sqrt{x} \]
To find \(x\), we can set the coefficients equal to each other:
\[ 19\sqrt{5} = 13\sqrt{x} \]
If we divide both sides by 13, we get:
\[ \sqrt{x} = \frac{19}{13} \sqrt{5} \]
Squaring both sides gives us:
\[ x = \left(\frac{19}{13}\right)^2 \cdot 5 \]
Calculating \(\left(\frac{19}{13}\right)^2\):
\[ \left(\frac{19}{13}\right)^2 = \frac{361}{169} \]
Now substituting back:
\[ x = \frac{361}{169} \cdot 5 = \frac{1805}{169} \]
Thus, the completed expression is:
\[ 10 \sqrt{5} + 3 \sqrt{45} = 19\sqrt{5} = 13\sqrt{\frac{1805}{169}} \]
In conclusion, the answer is:
\[ x = \frac{1805}{169} \]
So \(10\sqrt{5} + 3\sqrt{45} = 13\sqrt{\frac{1805}{169}}\).
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