Simplify, by rationalizing the Denominator (Problem Below)?

Question

I'd like to have a simple step-by-step run though of this below problem:

Sq Rt of 2X^3 .. Divided By .. Sq Rt of 8X^6

Answer 1

in the denominator you can separate the 8 into 2*4 and get the 4 out of the radical since it is a perfect square and simplifies to 2. That leaves a 2 under the radical. The x^6 can be thought of as x^2 * x^2 * x^2 which when sq rooted will simplify to x^3. So the only thing left in the denominator is 2x^3sqrt2.



Now for the numerator. The x^3 can be broken down into x^2 * x and take the square root of the x^2, getting just x. so the numerator is now x sqrt (2x).



As you said, you need to rationalize the denominator, The problem is the sqrt 2. Multiply num and denom. by sqrt 2. In the numerator, sqrt2 (you had) times this sqrt 2, gives just 2. Likewise in the denominator, you are left with 2x^3 * 2 which is 4x^3.

After rationalizing the denom. you get x sqrt(2x) / 4x^3. An x term cancels in num and denom leaving

sqrt(2x) / 4x^2

Answer 2

This problem simplifies to Sqrt [2x^3 / 8x^6]



This simplifies to Sqrt [1 / 4x^3] = 1 / Sqrt (4x^3) = 1 / 2x * sqrt(x)



Multiply numerator and denominator by sqrt(x) to rationalize the denominator. You get



Sqrt(x)/ 2x^2

Answer 3

I'm going to change the equation to all exponents:



(2x^3)^0.5 / (8x^6)^0.5

I'm going to separate it into two parts, the coefficients and the variables:

Coeffiecients:

(2^0.5) / (8^0.5)

=(2/8)^0.5 (If bot the numerator and denominator are raised to the same power, then you can change the equation so that you divide the numbers before you raise them to that power).

=(1/4)^0.5

=1/2



Now let's look at the variables:

(x^3)^0.5 / (x^6)^0.5

= (x^3/x^6)^0.5

= (1/x^3)^0.5

= (x^-3)^0.5 (an exponent in the denominator is the same as the negative of the exponent in the numerator)

=x^(-3/2)

=1/x(x^0.5) or 1/(x * sq rt x)



Putting the coefficients and variables back together yields:



(1/2) * (1/x^(3/2))

= 1/[2x^(3/2)] or 1/(2x*sqrt(x))

Answer 4

Nancy B has most of it right. The groupings was not correct in line 2, though.



"This simplifies to SQRT [1 / 4x^3] = 1 / SQRT (4x^3) = 1 / 2x * SQRT(x)"



The last expression should be: = 1 / (2x * SQRT(x))

Answer 5

power it



2x^3 / 8x^6 = 1 / 4x^3

Answer 6

2X^3/2 divded by 8X^6/2 now x^1 1/2 diveded by 4X^3

Answer 7

Radicals are actually not accepted interior the denominator, that's equivalent to an incorrect fraction. So: Multiply the two numerator and denominator by the denominator. ?3 (?3)(?15x) (?3)(?15x) 3?5x ?5x -------- = ------------------- = ---------------- = ---------- = --------- ?15x (?15x)(?15x) 15x 15x 5x the amazing answer is: D

Answer 8

sqrt(2*x^3)/sqrt(8*x^6) =(1/2)*sqrt(x^3)/sqrt(x^6) = (1/2)*x*sqrt(x)/x^3