SOLUTION: Simplify Completely, keep all answers and work in radical form 5*sqrt(128x^3)+4*sqrt(98x^2)-3x*sqrt(162x)

Algebra ->  Radicals -> SOLUTION: Simplify Completely, keep all answers and work in radical form 5*sqrt(128x^3)+4*sqrt(98x^2)-3x*sqrt(162x)      Log On


   

Question 1192821: Simplify Completely, keep all answers and work in radical form
5*sqrt(128x^3)+4*sqrt(98x^2)-3x*sqrt(162x)
Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

Notice that 128 = 64*2 showing that 64 is the largest perfect square factor of 128.
Similarly, x^2 is the largest perfect square factor of x^3

We can rewrite 128x^3 into 64x^2*2x to be able to perform this simplification
sqrt%28128x%5E3%29+=+sqrt%2864x%5E2%2A2x%29
sqrt%28128x%5E3%29+=+sqrt%2864x%5E2%29%2Asqrt%282x%29 Use the rule that sqrt(A*B) = sqrt(A)*sqrt(B)
sqrt%28128x%5E3%29+=+sqrt%28%288x%29%5E2%29%2Asqrt%282x%29
sqrt%28128x%5E3%29+=+8x%2Asqrt%282x%29
Such that x+%3E=+0

Through similar logic, we can also say:
sqrt%2898x%5E2%29+=+sqrt%2849x%5E2%2A2%29
sqrt%2898x%5E2%29+=+sqrt%2849x%5E2%29%2Asqrt%282%29
sqrt%2898x%5E2%29+=+sqrt%28%287x%29%5E2%29%2Asqrt%282%29
sqrt%2898x%5E2%29+=+7x%2Asqrt%282%29

And also,
sqrt%28162x%29+=+sqrt%2881%2A2x%29
sqrt%28162x%29+=+sqrt%2881%29%2Asqrt%282x%29
sqrt%28162x%29+=+9%2Asqrt%282x%29

Use the three results we got earlier above to simplify the given expresdsion like so
5%2Asqrt%28128x%5E3%29%2B4%2Asqrt%2898x%5E2%29-3x%2Asqrt%28162x%29

5%2A8x%2Asqrt%282x%29%2B4%2A7x%2Asqrt%282%29-3x%2A9%2Asqrt%282x%29

40x%2Asqrt%282x%29%2B28x%2Asqrt%282%29-27x%2Asqrt%282x%29

40x%2Asqrt%282x%29-27x%2Asqrt%282x%29%2B28x%2Asqrt%282%29

%2840x-27x%29%2Asqrt%282x%29%2B28x%2Asqrt%282%29

13x%2Asqrt%282x%29%2B28x%2Asqrt%282%29

Therefore,
5%2Asqrt%28128x%5E3%29%2B4%2Asqrt%2898x%5E2%29-3x%2Asqrt%28162x%29 simplifies to 13x%2Asqrt%282x%29%2B28x%2Asqrt%282%29

Or we can say

only when x+%3E=+0