I'm assuming you can handle long division and multiplication, without me writing it out, but I've avoided it where possible, by using factoring...
Start with the squared part:
59^(2/3) = (59^2)^1/3
((60-1)^2)^(1/3)
= (60^2 - 2*60*1^2)
=(3600 - 120 +1)^1/3
= 3481^(1/3)
So now we want the cube root of 3481
Look for perfect cubes near 3481:
15^3 = 225 * 15 = 3375, so that's pretty close.
To close in on it, you can do successive approximations, using halley's method. This says if you have a guess (a) about the cubed root of R, the following formula will give you a closer guess (b)
b = a(a^3 + 2R)/(2a^3 + R)
= 15(3375 + 3481*2)/(2*3375+3481)
= 15(3375 + 6962)/(6750+3481)
= 15(10337/10231)
= 15(1.010360669)
= 15.15541003 (this cubed gives you 3480.992, which is VERY close)
You can keep doing this till you get a value that cubes to 3481 exactly, but 15.155 seems close enough.
Comments
I'm assuming you can handle long division and multiplication, without me writing it out, but I've avoided it where possible, by using factoring...
Start with the squared part:
59^(2/3) = (59^2)^1/3
((60-1)^2)^(1/3)
= (60^2 - 2*60*1^2)
=(3600 - 120 +1)^1/3
= 3481^(1/3)
So now we want the cube root of 3481
Look for perfect cubes near 3481:
15^3 = 225 * 15 = 3375, so that's pretty close.
To close in on it, you can do successive approximations, using halley's method. This says if you have a guess (a) about the cubed root of R, the following formula will give you a closer guess (b)
b = a(a^3 + 2R)/(2a^3 + R)
= 15(3375 + 3481*2)/(2*3375+3481)
= 15(3375 + 6962)/(6750+3481)
= 15(10337/10231)
= 15(1.010360669)
= 15.15541003 (this cubed gives you 3480.992, which is VERY close)
You can keep doing this till you get a value that cubes to 3481 exactly, but 15.155 seems close enough.
The ^ (a.k.a caret) is just a sign that it's to the power of something. Like 2^8 would be 2 to the power of 8, which is 256.
well first you can square 59. and you get 3481. than u need to cube root it so u can simplify that