Simplify: $\sqrt{24 y^{11}}$
Answer (2)
Factor the number 24 into its prime factors.
Rewrite the expression as a product of perfect squares and remaining factors under the square root.
Take out the perfect squares from the square root.
Simplify the expression to obtain the final answer: 2 y 5 6 y
Explanation
Understanding the Problem We are asked to simplify the expression 24 y 11 . This involves simplifying a square root containing a numerical coefficient and a variable raised to a power.
Factoring the Expression First, we can factor the number 24 into its prime factors: 24 = 2 3 ⋅ 3 = 2 2 ⋅ 2 ⋅ 3 . We can also rewrite y 11 as y 10 ⋅ y .
Rewriting the Expression Now we rewrite the expression as 2 2 ⋅ 2 ⋅ 3 ⋅ y 10 ⋅ y .
Simplifying the Square Root We can take out the perfect squares from the square root. Since 2 2 = 2 and y 10 = y 5 , we have 2 y 5 2 ⋅ 3 ⋅ y .
Final Simplification Finally, we simplify the expression to 2 y 5 6 y .
Examples
Simplifying radical expressions is useful in various fields, such as physics and engineering, where you might need to calculate distances or areas involving square roots. For example, if you are calculating the length of the diagonal of a rectangle with sides involving radicals, you would need to simplify the resulting expression to get a more manageable form. This skill is also fundamental in more advanced mathematical topics like calculus and differential equations.
The expression 24 y 11 simplifies to 2 y 5 6 y by factoring 24 and rewriting the variable part. We extract the perfect squares and simplify the remaining factors under the square root. The final answer is 2 y 5 6 y .
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