Guerrero Norton
06/08/2023 · Junior High School
Solve by viing midpoint method \( (\sqrt{2}, 3 \sqrt{5})(\sqrt{2}-2 \sqrt{5}) \)
UpStudy ThothAI Solution
Tutor-Verified Answer
Step-by-step Solution
To solve the expression \( (\sqrt{2}, 3 \sqrt{5})(\sqrt{2}-2 \sqrt{5}) \) using the midpoint method, we first need to clarify the expression. It seems like you are trying to evaluate the product of two terms.
Let's denote:
- \( a = \sqrt{2} \)
- \( b = 3\sqrt{5} \)
- \( c = \sqrt{2} - 2\sqrt{5} \)
Now, we can rewrite the expression as:
\[
(a, b)(c) = a \cdot c + b \cdot c
\]
Calculating \( a \cdot c \):
\[
a \cdot c = \sqrt{2}(\sqrt{2} - 2\sqrt{5}) = \sqrt{2} \cdot \sqrt{2} - \sqrt{2} \cdot 2\sqrt{5} = 2 - 2\sqrt{10}
\]
Calculating \( b \cdot c \):
\[
b \cdot c = 3\sqrt{5}(\sqrt{2} - 2\sqrt{5}) = 3\sqrt{5} \cdot \sqrt{2} - 3\sqrt{5} \cdot 2\sqrt{5} = 3\sqrt{10} - 30
\]
Now, we can combine these results:
\[
(a, b)(c) = (2 - 2\sqrt{10}) + (3\sqrt{10} - 30)
\]
\[
= 2 - 30 + (-2\sqrt{10} + 3\sqrt{10}) = -28 + \sqrt{10}
\]
Thus, the final result of the expression \( (\sqrt{2}, 3 \sqrt{5})(\sqrt{2}-2 \sqrt{5}) \) is:
\[
\boxed{-28 + \sqrt{10}}
\]
Quick Answer
The result is \(-28 + \sqrt{10}\).
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