Simplify the radical expression $\sqrt[3]{125 x^2 y^7}$
Answer (2)
Rewrite the expression inside the cube root: 3 125 x 2 y 7 = 3 5 3 x 2 ( y 2 ) 3 y .
Take out terms with a power of 3 from the cube root.
Simplify the expression: 5 y 2 3 x 2 y .
The final simplified expression is 5 y 2 3 x 2 y .
Explanation
Understanding the Problem We are given the radical expression 3 125 x 2 y 7 and our goal is to simplify it.
Breaking Down the Expression First, let's break down the expression inside the cube root. We know that 125 = 5 3 , and we can rewrite y 7 as y 6 y , which is the same as ( y 2 ) 3 y . So, we have: 3 125 x 2 y 7 = 3 5 3 x 2 ( y 2 ) 3 y
Simplifying the Cube Root Now, we can take out the terms that have a power of 3 (or a multiple of 3) from the cube root: 3 5 3 x 2 ( y 2 ) 3 y = 5 y 2 3 x 2 y
Final Simplified Expression So, the simplified expression is 5 y 2 3 x 2 y .
Examples
Radical expressions are used in various fields, such as engineering and physics, to simplify complex formulas and calculations. For example, when calculating the period of a pendulum, you might encounter a radical expression that needs simplification to find the period more easily. Simplifying radicals helps in making these calculations more manageable and understandable, allowing for more efficient problem-solving in real-world applications.
The radical expression 3 125 x 2 y 7 simplifies to 5 y 2 3 x 2 y by factoring terms into perfect cubes and taking them out of the cube root.
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