Number of Decimal Places when square and cube rooting

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jsl: Master | Next Rank: 500 Posts; Posts: 203; Joined: Wed Jun 04, 2008 1:01 am; Location: Windsor; Thanked: 5 times; GMAT Score:650

by jsl » Thu Jun 19, 2008 12:22 pm

Hiya,

I keep getting confused about what happens when you square root or cube root a number. I saw the following question in a practise test and I don't consistently know how to calculate the answer. Can someone offer some tips so I can remember the rules?

square root ( cube root ( 0.000064 ) ) = ?

...the answer is 0.2 (I think).

How do I know how many places to shift decimals?

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Source: Beat The GMAT — Problem Solving | Thu Jun 19, 2008 12:22 pm

kishore: Senior | Next Rank: 100 Posts; Posts: 62; Joined: Fri Dec 21, 2007 2:58 pm; Location: Fremont , California; Thanked: 3 times

by kishore » Thu Jun 19, 2008 12:30 pm

square root ( cube root ( 0.000064 ) ) = ?

you can re-write 0.000064 = 64/1000000

= 2 ^ 6/ 10 ^ 6
= (2/10) ^ 6
= ((2/10)^ 2)^3)

There fore, square root ( cube root ( 0.000064 ) )

= sqrt(cbrt(2 ^ 6/ 10 ^ 6))
= sqrt (cbrt((2/10)^ 2)^3))
= 2/10
= 0.2

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Stuart@KaplanGMAT: GMAT Instructor; Posts: 3225; Joined: Tue Jan 08, 2008 2:40 pmFollowed by: 614 members; Location: Toronto; Thanked: 1710 times; GMAT Score:800

by Stuart@KaplanGMAT » Thu Jun 19, 2008 12:39 pm

For this specific question, it would probably be a lot faster to backsolve than to use algebra.

To backsolve, we'd square then cube an answer choice. For example:

(b) .2 ----> (.2)^2 = .04 -----> (.04)^3 = (.04)(.04)(.04) = .000064

.000064 matches the question, so (b) would be correct.

If (b) had given us a result less than .000064, we'd have looked for a bigger choice; if (b) had given us a result more than .000064, we'd have looked for a smaller choice (which, since the answers are arranged in ascending order, would mean that we could have just chosen (a)).

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pharmd: Newbie | Next Rank: 10 Posts; Posts: 9; Joined: Tue Mar 18, 2008 12:29 pm

Another Approach

by pharmd » Thu Jun 19, 2008 1:49 pm

What works for me is to count the numbers/ places of digits .. for instance in 0.000064 - there are 6 digits after the decimal point -> hence this can also be written as 64 X 10 ^ -6
4 x 4 x 4 = 64 therefore the cube of 64 x 10^-6 is 4 x 10 ^-2
Square root of this number is 2 x 10^-1 -> 0.2
I hope it helps ... let me know is you need any clarifications

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jsl: Master | Next Rank: 500 Posts; Posts: 203; Joined: Wed Jun 04, 2008 1:01 am; Location: Windsor; Thanked: 5 times; GMAT Score:650

Re: Another Approach

by jsl » Fri Jun 20, 2008 3:04 am

pharmd wrote:therefore the cube of 64 x 10^-6 is 4 x 10 ^-2

Hiya - thanks for the advice.... I understand how you got from 64 --> 4. But what I don't understand is how you got from 10^-6 to 10^-2 This is a difference of 4 decimal places but when you cube something, that would imply to me that you move 3 decimal places....

Is there an easy rule I can use to remember this?

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Ian Stewart: GMAT Instructor; Posts: 2623; Joined: Mon Jun 02, 2008 3:17 amFollowed by: 355 members; Location: Montreal; Thanked: 1090 times; GMAT Score:780

by Ian Stewart » Fri Jun 20, 2008 5:17 am

jsl,

You're certainly not alone if you find multiplying or dividing decimals confusing. Writing numbers as decimals is great if you have a calculator, but decimals are often very awkward if you just have pen and paper- as you do on the GMAT. Fractions, on the other hand, are easy to deal with, if you know the basic rules for multiplying and dividing fractions. For example:

1.125/0.625 = ?

That's a bit time consuming if you work with decimals. But:

(9/8)/(5/8) = 9/5 = 1.8

is very fast. For the question you've posted, I'd certainly use kishore's approach. I'd convert the decimal to the fraction 2^6/10^6. Then, to cube root, we divide the exponents by 3. To take the positive square root, we divide again by 2. We're left with 2/10 = 0.2.

As for your question about cube rooting, note that taking the cube root of a number is the same as raising the number to the exponent (1/3). So, if we have, for example, the cube root of x^6, that's equal to:

(x^6)^(1/3)

and because of the 'tower of powers' rule, one of the basic (and most important!) powers rules, we multiply the powers here. So this is just equal to x^2. That's why the cube root of (2/10)^6 is just (2/10)^2.

For online GMAT math tutoring, or to buy my higher-level Quant books and problem sets, contact me at ianstewartgmat at gmail.com

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