Question - Simplifying Radical Expressions
mathsimplify radical expressionsradical expression simplificationsquare root functionperfect squares in radicals
Solution:
To simplify the radical expression $$ \sqrt{48x^{13}y^9} $$, you'll need to break down the numbers and variables inside the radical into factors that are perfect squares, as the square root function will cancel with squared terms.First, factor the number 48 and express it as a product of squared numbers:48 = 16 * 3, where 16 is a perfect square (since 16 = 4^2).Next, for the variable terms $$ x^{13} $$ and $$ y^9 $$, we want to find the highest even powers since they can be taken out of the square root as a single power. For $$ x^{13} $$, the highest even power less than 13 is 12, which can be written as $$ (x^6)^2 $$. For $$ y^9 $$, the highest even power less than 9 is 8, which can be written as $$ (y^4)^2 $$.The expression becomes:$$ \sqrt{16*3*(x^6)^2*(y^4)^2} $$Taking the square roots of the perfect squares, we get:$$ 4x^6y^4\sqrt{3} $$So the simplified form of the radical expression $$ \sqrt{48x^{13}y^9} $$ is $$ 4x^6y^4\sqrt{3} $$.
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