Simplify $2\sqrt{125v}$
Understand the Problem
The question asks to simplify the expression $2\sqrt{125v}$. This involves simplifying the square root by factoring out perfect squares from 125.
Answer
$10\sqrt{5v}$
Answer for screen readers
$10\sqrt{5v}$
Steps to Solve
- Factor 125
Find the prime factorization of 125. We see that $125 = 5 \cdot 25 = 5 \cdot 5 \cdot 5 = 5^3$. We can rewrite the expression as:
$2\sqrt{5^3 v}$
- Rewrite the square root using perfect squares
Rewrite $5^3$ as $5^2 \cdot 5$ to identify the perfect square. Separate out the perfect square from the square root
$2\sqrt{5^2 \cdot 5 \cdot v}$ $2\sqrt{5^2} \cdot \sqrt{5v}$
- Simplify the square root
Simplify $\sqrt{5^2}$ to 5, then multiply by the 2 outside the square root. $2 \cdot 5 \sqrt{5v}$ $10\sqrt{5v}$
$10\sqrt{5v}$
More Information
The expression $2\sqrt{125v}$ simplifies to $10\sqrt{5v}$. We extracted the perfect square factor from 125 and simplified the radical.
Tips
A common mistake is not fully factoring the number under the square root. For example, one might stop at $2\sqrt{25 \cdot 5v}$ and not take the square root of 25. Also, students might forget to multiply the simplified square root by the coefficient that is already outside the square root.
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