Simplify square root of 2n^3
Brief Explanation: The question asks to perform a mathematical simplification on the expression given, which includes the square root of a product of 2 and a variable n raised to the third power. The result should be an expression where the square root is simplified as much as possible, possibly by factoring out perfect squares and applying the properties of square roots to the variables involved.
$\sqrt{2 n^{3}}$
Answer
Solution:
Step 1:
Express $2n^3$ in terms of $n^2$ by writing it as $n^2 \cdot (2n)$.
Step 1.1:
Extract $n^2$ from under the radical sign: $\sqrt{2(n^2 \cdot n)}$.
Step 1.2:
Rearrange the terms inside the radical: $\sqrt{n^2 \cdot 2n}$.
Step 1.3:
Enclose $2n$ within parentheses: $\sqrt{n^2 \cdot (2n)}$.
Step 2:
Remove the perfect square $n^2$ from under the square root to get: $n\sqrt{2n}$.
Knowledge Notes:
To simplify the square root of a product that includes a perfect square, you can use the property of square roots that states $\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$, where $a$ and $b$ are non-negative real numbers. In this case, if $a$ is a perfect square, then $\sqrt{a}$ is an integer.
The given expression is $\sqrt{2n^3}$. The goal is to simplify it by factoring out the perfect square from under the radical. The steps involve:
Factoring the expression under the radical to identify the perfect square ($n^2$ in this case).
Using the property of square roots to separate the perfect square from the rest of the expression.
Simplifying the expression by taking the square root of the perfect square, which yields an integer.
Writing the final simplified expression as the product of the integer obtained from the perfect square and the square root of the remaining expression.
In the given problem, $n^2$ is the perfect square that can be factored out from $2n^3$, resulting in the simplified form $n\sqrt{2n}$.
