2√5(3√5 + √2),
Understand the Problem
The question involves evaluating a mathematical expression that includes square roots and parentheses. We will simplify and solve the expression step by step.
Answer
The final answer is \(30 + 2\sqrt{10}\).
Answer for screen readers
The final answer is (30 + 2\sqrt{10}).
Steps to Solve
- Distribute the square root term We begin by distributing $2\sqrt{5}$ through the terms inside the parentheses:
$$ 2\sqrt{5}(3\sqrt{5} + \sqrt{2}) = 2\sqrt{5} \cdot 3\sqrt{5} + 2\sqrt{5} \cdot \sqrt{2} $$
- Multiply the terms Next, we multiply the terms:
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For the first term: $$ 2\sqrt{5} \cdot 3\sqrt{5} = 2 \cdot 3 \cdot (\sqrt{5} \cdot \sqrt{5}) = 6 \cdot 5 = 30 $$
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For the second term: $$ 2\sqrt{5} \cdot \sqrt{2} = 2 \cdot \sqrt{10} $$
- Combine the results Now combine the results from both multiplications:
$$ 30 + 2\sqrt{10} $$
The final answer is (30 + 2\sqrt{10}).
More Information
This problem showcases how to simplify an expression using the distributive property. Working with square roots requires careful handling, especially when multiplying and combining like terms.
Tips
- Forgetting to distribute: A common mistake is skipping the distribution step. Make sure to distribute carefully.
- Incorrectly multiplying square roots: Misapplying the square root multiplication can lead to errors. Always remember that $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$.
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