Simplify $4\sqrt{3}(5\sqrt{8}+2\sqrt{3})$
Understand the Problem
The question asks to simplify a mathematical expression involving radicals. The expression is $4\sqrt{3}(5\sqrt{8}+2\sqrt{3})$. We need to distribute and simplify the radicals to get the final answer.
Answer
$24 + 40\sqrt{6}$
Answer for screen readers
$24 + 40\sqrt{6}$
Steps to Solve
- Distribute $4\sqrt{3}$ to both terms inside the parentheses
Apply the distributive property: $4\sqrt{3}(5\sqrt{8}+2\sqrt{3}) = 4\sqrt{3} \cdot 5\sqrt{8} + 4\sqrt{3} \cdot 2\sqrt{3}$
- Multiply the coefficients and the radicals separately for each term
$4\sqrt{3} \cdot 5\sqrt{8} = (4 \cdot 5)(\sqrt{3} \cdot \sqrt{8}) = 20\sqrt{24}$ $4\sqrt{3} \cdot 2\sqrt{3} = (4 \cdot 2)(\sqrt{3} \cdot \sqrt{3}) = 8\sqrt{9}$
- Simplify the radicals
$\sqrt{24} = \sqrt{4 \cdot 6} = \sqrt{4} \cdot \sqrt{6} = 2\sqrt{6}$ $\sqrt{9} = 3$
- Substitute the simplified radicals back into the expression
$20\sqrt{24} + 8\sqrt{9} = 20(2\sqrt{6}) + 8(3)$
- Perform the multiplications
$20(2\sqrt{6}) = 40\sqrt{6}$ $8(3) = 24$
- Combine the terms
$40\sqrt{6} + 24$
$24 + 40\sqrt{6}$
More Information
The simplified form of the expression $4\sqrt{3}(5\sqrt{8}+2\sqrt{3})$ is $24 + 40\sqrt{6}$.
Tips
A common mistake is to forget to distribute the $4\sqrt{3}$ to both terms inside the parenthesis. Also, simplification of radicals can be tricky and might lead to errors. Make sure to factor out perfect squares from the radicals to simplify them correctly.
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