Simplify √12 + √27
Understand the Problem
The question asks to simplify the expression which involves adding two square roots: √12 + √27. We need to simplify radicals and combine like terms.
Answer
$5\sqrt{3}$
Answer for screen readers
$5\sqrt{3}$
Steps to Solve
- Simplify $\sqrt{12}$
Find the prime factorization of 12: $12 = 2 \times 2 \times 3 = 2^2 \times 3$. Therefore, $\sqrt{12} = \sqrt{2^2 \times 3} = \sqrt{2^2} \times \sqrt{3} = 2\sqrt{3}$.
- Simplify $\sqrt{27}$
Find the prime factorization of 27: $27 = 3 \times 3 \times 3 = 3^2 \times 3$. Therefore, $\sqrt{27} = \sqrt{3^2 \times 3} = \sqrt{3^2} \times \sqrt{3} = 3\sqrt{3}$.
- Combine the simplified radicals
Now we have $2\sqrt{3} + 3\sqrt{3}$. Since they both have the same radical part, $\sqrt{3}$, we can add them like regular terms: $2\sqrt{3} + 3\sqrt{3} = (2+3)\sqrt{3} = 5\sqrt{3}$.
$5\sqrt{3}$
More Information
The simplified expression of $\sqrt{12} + \sqrt{27}$ is $5\sqrt{3}$. This is obtained by simplifying each square root individually and then combining them.
Tips
A common mistake is to try to add the numbers inside the square roots directly, which is incorrect. For example, doing $\sqrt{12 + 27} = \sqrt{39}$ is wrong. You must simplify each radical first before adding. Another mistake is not knowing the perfect squares.
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