Simplify $4\sqrt{3}(5\sqrt{8}+2\sqrt{3})$.
Understand the Problem
The question is asking to simplify the expression $4\sqrt{3}(5\sqrt{8} + 2\sqrt{3})$. This involves distributing the $4\sqrt{3}$ across the terms inside the parentheses and then simplifying the resulting terms by combining like radicals.
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: $4\sqrt{3} * (5\sqrt{8}) + 4\sqrt{3} * (2\sqrt{3})$
- Multiply the coefficients and the radicals in the first term
$4\sqrt{3} * 5\sqrt{8} = (45) * (\sqrt{3}\sqrt{8}) = 20\sqrt{24}$
- Multiply the coefficients and the radicals in the second term
$4\sqrt{3} * 2\sqrt{3} = (42) * (\sqrt{3}\sqrt{3}) = 8 * 3 = 24$
- Simplify $\sqrt{24}$
Find the prime factorization of 24: $24 = 222*3 = 2^3 * 3$. Thus, $\sqrt{24} = \sqrt{2^2 * 2 * 3} = 2\sqrt{6}$
- Substitute the simplified radical back into the first term
$20\sqrt{24} = 20 * 2\sqrt{6} = 40\sqrt{6}$
- Combine the simplified terms
$40\sqrt{6} + 24$
$24 + 40\sqrt{6}$
More Information
The simplified expression is $24 + 40\sqrt{6}$. It is in its simplest form because $\sqrt{6}$ cannot be simplified further and there are no like terms to combine.
Tips
A common mistake is not simplifying $\sqrt{8}$ and $\sqrt{24}$ completely, which can lead to incorrect final answers. Another common mistake is incorrectly multiplying the coefficients or the radicals during the distribution step. For example, some students might incorrectly multiply $4\sqrt{3} * 5\sqrt{8}$ as $9\sqrt{24}$ or $20\sqrt{11}$.
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