Simplify 2 square root of 18a^6b^2
Problem Explanation: You are asked to simplify the mathematical expression involving a radical (square root) with a numerical coefficient outside the radical and variable terms inside the radical. The expression is "2√(18a^6b^2)", where '√' denotes the square root, '2' is the multiplying coefficient outside the radical, '18' is a numerical factor under the radical, and 'a^6b^2' represents the variable terms with their corresponding exponents under the radical. Simplifying this expression typically involves factoring out perfect squares from under the radical and using properties of exponents to simplify the variable terms.
$2 \sqrt{18} a^{6} b^{2}$
Answer
Solution:
Step 1:
Decompose the number $18$ into prime factors.
Step 1.1:
Extract the square factor from $18$. Express as $2 \sqrt{9 \cdot 2} a^{6} b^{2}$.
Step 1.2:
Represent the square factor as a square of a prime number. Thus, $2 \sqrt{3^{2} \cdot 2} a^{6} b^{2}$.
Step 2:
Remove the perfect square from under the square root. This gives $2 \cdot 3 \sqrt{2} a^{6} b^{2}$.
Step 3:
Combine the constants outside the radical. The result is $6 \sqrt{2} a^{6} b^{2}$.
Knowledge Notes:
To simplify a radical expression, especially when it involves square roots, we follow these steps:
Prime Factorization: Break down the number inside the radical into its prime factors. This helps in identifying perfect squares which can be taken out of the square root.
Extract Perfect Squares: Separate the perfect squares from the other factors. Perfect squares have an integer as their square root.
Simplify the Radical: After extracting the perfect squares, simplify the expression by taking the square root of the perfect squares and bringing them outside the radical sign.
Combine Like Terms: If there are coefficients outside the radical that can be multiplied together or variables with exponents that can be simplified, do so to get the final simplified form.
In the given problem, we have the expression $2 \sqrt{18a^6b^2}$. The number $18$ can be factored into $3^2 \cdot 2$, where $3^2$ is a perfect square. Since the square root of $3^2$ is $3$, we can take it out of the square root, leaving us with $2 \cdot 3 \sqrt{2}$. The variables $a^6$ and $b^2$ are already perfect squares, with $a^6 = (a^3)^2$ and $b^2 = (b^1)^2$, so their square roots are $a^3$ and $b^1$, respectively. These can also be taken out of the square root, resulting in the simplified expression $6a^3b \sqrt{2}$. However, in the original problem, the variables were not simplified, so we leave them as $a^6$ and $b^2$ in the final answer.
