Simplify [tex]$\frac{\sqrt[3]{2 b}}{\sqrt[3]{5 a^7}}$[/tex]. Assume that all variables represent positive numbers.
Answer (1)
Combine the cube roots into a single expression: 3 5 a 7 3 2 b = 3 5 a 7 2 b .
Factor out a 6 from the denominator: 3 5 a 7 2 b = 3 5 a 6 a 2 b .
Take a 2 out of the cube root: a 2 1 3 5 a 2 b .
Rationalize the denominator: a 2 1 3 5 a ⋅ 25 a 2 2 b ⋅ 25 a 2 = 5 a 3 3 50 a 2 b .
The simplified expression is 5 a 3 3 50 a 2 b .
Explanation
Understanding the Problem We are asked to simplify the expression 3 5 a 7 3 2 b , assuming that all variables represent positive numbers.
Combining Cube Roots We can rewrite the expression as a single cube root: 3 5 a 7 3 2 b = 3 5 a 7 2 b .
Factoring the Denominator Next, we want to factor out a 6 from the denominator inside the cube root, since a 6 is a perfect cube: 3 5 a 7 2 b = 3 5 a 6 a 2 b .
Simplifying the Expression Now we take a 2 out of the cube root: 3 5 a 6 a 2 b = a 2 1 3 5 a 2 b .
Rationalizing the Denominator To rationalize the denominator, we multiply the numerator and denominator inside the cube root by ( 5 a ) 2 = 25 a 2 : a 2 1 3 5 a 2 b = a 2 1 3 5 a ⋅ 25 a 2 2 b ⋅ 25 a 2 = a 2 1 3 125 a 3 50 a 2 b .
Further Simplification Now we take 5 a out of the cube root in the denominator: a 2 1 3 125 a 3 50 a 2 b = a 2 1 5 a 3 50 a 2 b .
Final Result Finally, we combine the terms outside the cube root: a 2 1 5 a 3 50 a 2 b = 5 a 3 3 50 a 2 b .
Conclusion Therefore, the simplified expression is 5 a 3 3 50 a 2 b .
Examples
Imagine you are designing a container, and its volume is given by 3 2 b . However, due to manufacturing constraints, the actual volume is 3 5 a 7 . Simplifying the expression 3 5 a 7 3 2 b helps you understand the ratio between the designed and actual volumes, allowing you to adjust the design or manufacturing process to match the intended specifications. This kind of simplification is useful in engineering and design to optimize resource usage and ensure accuracy.
