SOLUTION: my child needs to do the cube root of 6561 without a calculator. but need to show steps!!!! please help!!!!

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Question 596147: my child needs to do the cube root of 6561 without a calculator. but need to show steps!!!!
please help!!!!
Answer by jsmallt9(3758) (Show Source):
You can put this solution on YOUR website!
Simplifying roots of any kind, without a calculator, involves factoring the radicand in a certain way. ("Radicand" is the name for the expression inside the radical.) For square roots you look for perfect square factors, for cube roots you look for perfect cube factors, etc.)

Since you are trying to simplify a cube root, you look for perfect cube factors. (Note: Although 1 is a perfect cube, factoring it out does us no good so we skip that perfect cube.) 2 cubed is 8. And 8 cannot be a factor of an odd number. (Divide it out with long division if you're not sure of this.) 3 cubed is 27. If you divide 6561 by 27 you will see that it does go into 6561 243 times. So 6561 = 27*243 and we can rewrite our cube root as:

Now we see if there is a perfect cube factor of 243. 2 cubed didn't work before and it won't now, either. Next we try 27 again. And it turns out that 243 = 27*9. Rewriting our cube root again we get:

Now we look for perfect cube factors of 9. But 8 still doesn't work and the other perfect cubes are larger than 9. So we are finished finding perfect cube factors.

Next we use a property of all radicals, , to separate each factor into its own cube root:

Next we simplify the cube roots of the perfect cubes:

Last of all we simplify anything that is outside the radical. Multiplying 3*3 we get:

So in its simplest form is .