Simplify. [tex]$\sqrt{10 v^2 y^4} \sqrt{2 v^7 y}$[/tex]
Answer (2)
Combine the square roots: 10 v 2 y 4 2 v 7 y = ( 10 v 2 y 4 ) ( 2 v 7 y ) .
Simplify the expression inside: ( 10 v 2 y 4 ) ( 2 v 7 y ) = 20 v 9 y 5 .
Factor out perfect squares and simplify: 20 v 9 y 5 = 4 v 8 y 4 ⋅ 5 v y .
Simplify the square root to get the final answer: 2 v 4 y 2 5 v y .
Explanation
Understanding the Problem We are given the expression 10 v 2 y 4 2 v 7 y to simplify. We assume that v and y are non-negative. We can combine the square roots and simplify the expression inside.
Objective We want to simplify the expression 10 v 2 y 4 2 v 7 y .
Combining Square Roots Combine the square roots: 10 v 2 y 4 2 v 7 y = ( 10 v 2 y 4 ) ( 2 v 7 y ) .
Simplifying Inside the Square Root Simplify the expression inside the square root: ( 10 v 2 y 4 ) ( 2 v 7 y ) = 20 v 2 + 7 y 4 + 1 = 20 v 9 y 5 .
Rewriting the Expression Rewrite the expression as 20 v 9 y 5 .
Factoring Perfect Squares Factor out perfect squares: 20 = 4 ⋅ 5 , v 9 = v 8 ⋅ v , y 5 = y 4 ⋅ y .
Rewriting with Factored Squares Rewrite the expression as 4 v 8 y 4 ⋅ 5 v y .
Simplifying the Square Root Simplify the square root: 4 v 8 y 4 ⋅ 5 v y = 4 v 8 y 4 5 v y = 2 v 4 y 2 5 v y .
Final Answer The simplified expression is 2 v 4 y 2 5 v y .
Examples
Square roots appear in many contexts, such as calculating distances or areas. For example, if you are designing a garden and need to calculate the length of a diagonal path across a rectangular plot, you would use the Pythagorean theorem, which involves square roots. Simplifying expressions with square roots, like the one in this problem, helps in making these calculations easier and more manageable. Understanding how to manipulate and simplify these expressions is a fundamental skill in various fields, including engineering, physics, and computer graphics.
To simplify the expression 10 v 2 y 4 2 v 7 y , we combine the square roots, simplify the contents, and factor out perfect squares. The final result is 2 v 4 y 2 5 v y .
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