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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Number of Decimal Places when square and cube rooting
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kishore
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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
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
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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)).
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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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
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
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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....pharmd wrote:therefore the cube of 64 x 10^-6 is 4 x 10 ^-2
Is there an easy rule I can use to remember this?
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Ian Stewart
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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.
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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