Simplify. [tex]$\sqrt{15 y^5 v^2} \sqrt{5 y^9 v}$[/tex]
Assume that all variables represent positive [tex]$r \in$[/tex]
Answer (2)
Combine the square roots: 15 y 5 v 2 5 y 9 v = ( 15 y 5 v 2 ) ( 5 y 9 v ) .
Simplify the product inside the square root: ( 15 y 5 v 2 ) ( 5 y 9 v ) = 75 y 14 v 3 .
Simplify the square root: 75 y 14 v 3 = 25 y 14 v 2 ⋅ 3 v .
Take the square root of perfect squares: 5 y 7 v 3 v .
5 y 7 v 3 v
Explanation
Understanding the Problem We are asked to simplify the expression 15 y 5 v 2 5 y 9 v , assuming that all variables represent positive real numbers. This means we can manipulate the expression using the properties of square roots and exponents without worrying about negative values or undefined results.
Combining Square Roots First, we combine the two square roots into a single square root by multiplying the terms inside: 15 y 5 v 2 5 y 9 v = ( 15 y 5 v 2 ) ( 5 y 9 v )
Simplifying the Product Next, we simplify the product inside the square root by multiplying the coefficients and adding the exponents of like variables: ( 15 y 5 v 2 ) ( 5 y 9 v ) = 15 ⋅ 5 ⋅ y 5 ⋅ y 9 ⋅ v 2 ⋅ v = 75 y 5 + 9 v 2 + 1 = 75 y 14 v 3 So now we have 75 y 14 v 3 .
Factoring and Rewriting Now, we want to simplify the square root. We can factor 75 as 25 ⋅ 3 , so the expression becomes 25 ⋅ 3 y 14 v 3 . We can rewrite this as 25 y 14 v 2 ⋅ 3 v .
Taking Square Roots We take the square root of the perfect squares: 25 y 14 v 2 = 5 y 7 v . The remaining term inside the square root is 3 v .
Final Simplified Expression Therefore, the simplified expression is 5 y 7 v 3 v .
Examples
Square roots and exponents are used extensively in physics, engineering, and computer graphics. For example, when calculating the distance between two points in space, you use the square root of the sum of the squares of the differences in their coordinates. Simplifying expressions with square roots and exponents makes these calculations easier and more efficient. Imagine you're designing a bridge and need to calculate the tension in a cable. The formula might involve square roots of expressions containing variables raised to different powers. Simplifying these expressions allows engineers to quickly and accurately determine the necessary specifications for the bridge's cables, ensuring its safety and stability.
The simplified expression for 15 y 5 v 2 5 y 9 v is 5 y 7 v 3 v . This is achieved by combining the square roots, simplifying the product, and extracting perfect squares. Careful attention to exponents and coefficients allows for accurate simplification.
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