Simplify 3 square root of 81x^16y^12
The problem is asking for the simplification of the mathematical expression which contains a numerical coefficient, a square root, and variables raised to even powers. To simplify the expression, you'll need to apply the properties of square roots and exponents to both the numeric and variable parts of the expression inside the square root. The task involves extracting factors from inside the square root that have perfect squares, thus simplifying the overall expression into a more elementary or reduced form without a square root or with a simplified radicand (the number under the square root symbol).
$3 \sqrt{81 x^{16} y^{12}}$
Answer
Solution:
Step 1:
Express $81 x^{16} y^{12}$ as $(9 x^{8} y^{6})^{2}$. Therefore, the expression becomes $3 \sqrt{(9 x^{8} y^{6})^{2}}$.
Step 2:
Extract the square root by taking out the terms from under the radical, assuming all variables represent non-negative numbers. This gives us $3 (9 x^{8} y^{6})$.
Step 3:
Combine the constants by multiplying $3$ with $9$. The simplified result is $27 x^{8} y^{6}$.
Knowledge Notes:
When simplifying expressions involving square roots, it's important to understand the properties of exponents and radicals. Here are the relevant knowledge points:
Square Root Property: The square root of a number squared is the absolute value of the original number. For example, $\sqrt{a^2} = |a|$.
Exponent Rules: When an exponent is raised to another exponent, you multiply the exponents. For example, $(a^m)^n = a^{m \cdot n}$.
Simplifying Radicals: To simplify a radical, you can factor the number or expression inside the radical into its prime factors or squares of factors and then take the square root of those squares.
Multiplication of Constants: Constants outside the radical can be multiplied by each other. For example, $a \cdot b = ab$.
Variables under the Radical: When dealing with variables under a radical, the same rules apply as with numerical expressions. If the variable is raised to an even power, it can be taken out of the radical as half of that power.
In the given problem, we used these principles to simplify the expression. We recognized that $81$ is a perfect square ($9^2$) and that the even powers of $x$ and $y$ can be simplified when taken out of the square root. This allowed us to simplify the expression without needing to calculate any actual square roots.
