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# Which of the following is the value of root{3rd rt{0,000064}

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Which of the following is the value of root{3rd rt{0,000064} [#permalink]

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30 Jul 2012, 02:02
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Which of the following is the value of $$\sqrt{\sqrt[3]{0.000064}}$$

(A) 0.004
(B) 0.008
(C) 0.02
(D) 0.04
(E) 0.2

Practice Questions
Question: 9
Page: 153
Difficulty: 650
[Reveal] Spoiler: OA

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Re: Which of the following is the value of root{3rd rt{0,000064} [#permalink]

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30 Jul 2012, 02:02
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SOLUTION

Which of the following is the value of $$\sqrt{\sqrt[3]{0.000064}}$$

(A) 0.004
(B) 0.008
(C) 0.02
(D) 0.04
(E) 0.2

There are many ways to solve this problem, for example:

$$\sqrt{\sqrt[3]{0.000064}}=\sqrt[6]{0.000064}=\sqrt[6]{\frac{2^6}{10^6}}=\frac{2}{10}=0.2$$.

For theory on exponents and roots check:
math-number-theory-88376.html

For practice on exponents and roots check:
tough-and-tricky-exponents-and-roots-questions-125967.html (DS problems)
tough-and-tricky-exponents-and-roots-questions-125956.html (PS problems)
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Re: Which of the following is the value of root{3rd rt{0,000064} [#permalink]

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30 Jul 2012, 02:39
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(E) .02

For simplicity
0.000064 can be $$\frac{64}{1000000}$$

Its cube root would be $$\frac{\sqrt[3]{64}}{\sqrt[3]{1000,000}}$$
i.e. $$\frac{4}{100}$$

its square root would be $$\frac{\sqrt[2]{4}}{\sqrt[2]{100}}$$
i.e. $$\frac{2}{10}$$

i.e. 0.2

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Re: Which of the following is the value of root{3rd rt{0,000064} [#permalink]

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30 Jul 2012, 03:04
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E it is...

= [(64/1000000)^1/3]^1/2
= [(4^3/10^6)^1/3]^1/2
= [ 4/10^2]^1/2
= 2/10
=0.2

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Re: Which of the following is the value of root{3rd rt{0,000064} [#permalink]

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03 Aug 2012, 04:59
Expert's post
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SOLUTION

Which of the following is the value of $$\sqrt{\sqrt[3]{0.000064}}$$

(A) 0.004
(B) 0.008
(C) 0.02
(D) 0.04
(E) 0.2

There are many ways to solve this problem, for example:

$$\sqrt{\sqrt[3]{0.000064}}=\sqrt[6]{0.000064}=\sqrt[6]{\frac{2^6}{10^6}}=\frac{2}{10}=0.2$$.

For theory on exponents and roots check:
math-number-theory-88376.html

For practice on exponents and roots check:
tough-and-tricky-exponents-and-roots-questions-125967.html (DS problems)
tough-and-tricky-exponents-and-roots-questions-125956.html (PS problems)
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Re: Which of the following is the value of root{3rd rt{0,000064} [#permalink]

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05 Jun 2013, 04:28
Bumping for review and further discussion*. Get a kudos point for an alternative solution!

*New project from GMAT Club!!! Check HERE

Theory on roots problems: math-number-theory-88376.html

All DS roots problems to practice: search.php?search_id=tag&tag_id=49
All PS roots problems to practice: search.php?search_id=tag&tag_id=113

Tough and tricky exponents and roots questions (DS): tough-and-tricky-exponents-and-roots-questions-125967.html
Tough and tricky exponents and roots questions (PS): new-tough-and-tricky-exponents-and-roots-questions-125956.html

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Re: Which of the following is the value of root{3rd rt{0,000064} [#permalink]

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05 Jun 2013, 09:06
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Which of the following is the value of $$\sqrt{\sqrt[3]{0.000064}}$$

(A) 0.004
(B) 0.008
(C) 0.02
(D) 0.04
(E) 0.2

If you know how to do calculations as is, then it is:
$$\sqrt{\sqrt[3]{0.000064}} = \sqrt{0.04} = 0.2$$

or

If you know that $$\sqrt{\sqrt[3]{0.000064}} = \sqrt[6]{0.000064}$$ and $$2^6 = 64$$
Then you can eliminate A, B, and D. Then just move decimals from C and E. C would be 12 decimal places which is WAY too much. 0.2 is 6 which is exactly what you want

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Re: Which of the following is the value of root{3rd rt{0,000064} [#permalink]

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05 Jun 2013, 18:09
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The above question can be re written as ((10^-6)^1/3)^1/2 * ((64^1/3)^1/2)

= (10^-6)^1/6 * (2^6)^1/6

= 10^1 * 2^1 = 0.2

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Re: Which of the following is the value of root{3rd rt{0,000064} [#permalink]

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02 Feb 2014, 23:19
Could I just clarify something with a solution to this problem? The explanation in the book was given as: 64 * 10^-6 under the square root and cubed signs (I can't figure out how to write them in). Then from there it goes down to square root of 4 * 10^-2 then 2 * 10^-1 which then equals to .2 which is the answer. I think I've figured out how they got from beginning to end by reading the Number Theory post mentioned at the top, but just need clarification. When you have a base to a fraction power like a^n/m that turns to into n root sign a^m correct? So in this case where you have cubed 10^-6 it would reverse to 10^-6/3 which can be reduced to 10^-2 right? I think that's how the book answer went about getting to .2 but would just like clarification. Also this strategy can be used with any large decimaled number that is under a root or cube sign right?

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Re: Which of the following is the value of root{3rd rt{0,000064} [#permalink]

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04 Feb 2014, 02:43
amjet12 wrote:
Could I just clarify something with a solution to this problem? The explanation in the book was given as: 64 * 10^-6 under the square root and cubed signs (I can't figure out how to write them in). Then from there it goes down to square root of 4 * 10^-2 then 2 * 10^-1 which then equals to .2 which is the answer. I think I've figured out how they got from beginning to end by reading the Number Theory post mentioned at the top, but just need clarification. When you have a base to a fraction power like a^n/m that turns to into n root sign a^m correct? So in this case where you have cubed 10^-6 it would reverse to 10^-6/3 which can be reduced to 10^-2 right? I think that's how the book answer went about getting to .2 but would just like clarification. Also this strategy can be used with any large decimaled number that is under a root or cube sign right?

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Re: Which of the following is the value of root{3rd rt{0,000064} [#permalink]

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22 May 2014, 12:20
(64/1000000)^1/6={2^(6)/10^(6)}^1/6 = {2^(6*1/6)/10^(6*1/6)}= 2/10= .2
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Re: Which of the following is the value of root{3rd rt{0,000064} [#permalink]

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19 Aug 2014, 17:16
This is how I tried it the second time - As long as one is able to find the final exponent on the 10 part of the value, one can simply pick the choice that matches that exponent value. In this case 10^(-6)*(1/6) = 1/10 = 0.1 and the only choice that has one decimal place is E.

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Re: Which of the following is the value of root{3rd rt{0,000064} [#permalink]

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19 Aug 2014, 22:04
Bunuel wrote:
Which of the following is the value of $$\sqrt{\sqrt[3]{0.000064}}$$

(A) 0.004
(B) 0.008
(C) 0.02
(D) 0.04
(E) 0.2

Practice Questions
Question: 9
Page: 153
Difficulty: 650

$$\sqrt{\sqrt[3]{0.000064}} = \sqrt[(2*3)]{\frac{64}{10^6}} = (\frac{2^6}{10^6})^{\frac{1}{6}} = \frac{2}{10} = 0.2$$

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Re: Which of the following is the value of root{3rd rt{0,000064} [#permalink]

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09 Dec 2014, 20:26
cube root of 64 is 4. because we need the cube root of 0.000064, 4 needs to have (1/3) as many digits to the right of the decimal as 64, so 0.04.

Similarly, we need the square root of 0.04, so the square root of 4 is 2, but we need half the digits to the right of the decimal, so 0.2

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Re: Which of the following is the value of root{3rd rt{0,000064} [#permalink]

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27 Apr 2015, 12:39
--> (0,2^6)^1/6 = 0,2
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Re: Which of the following is the value of root{3rd rt{0,000064} [#permalink]

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11 Apr 2016, 16:30
Attached is a visual that should help.
Attachments

Screen Shot 2016-04-11 at 4.29.25 PM.png [ 148.45 KiB | Viewed 7685 times ]

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Re: Which of the following is the value of root{3rd rt{0,000064} [#permalink]

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03 May 2016, 08:55
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Bunuel wrote:
Which of the following is the value of $$\sqrt{\sqrt[3]{0.000064}}$$

(A) 0.004
(B) 0.008
(C) 0.02
(D) 0.04
(E) 0.2

Solution:

Let's review the notation first. When an exponent is a fraction, that exponent indicates taking a root. So if we have, for example, 27^1/3, the 1/3 instructs us to take the cube root of 27, which is 3. Similarly, if the exponent were 1/2, such as in 25^1/2, the 1/2 instructs us to take the square root of 25, which is 5.

To solve this question, we can refer to two rules:

1) If a decimal with a finite number of decimal places is a perfect cube, its cube root will have exactly one-third of the number of decimal places. Thus, a perfect cube decimal must have a number of decimal places that is a multiple of 3.

2) If a decimal with a finite number of decimal places is a perfect square, its square root will have exactly half of the number of decimal places. Thus, a perfect square decimal must have an even number of decimal places.

Let's look first at (0.000064)^1/3. The 1/3 instructs us to take the cube root of 0.000064. By rule number 1, the cube root of 0.000064 = 0.04. We were able obtain this value because 0.000064 has 6 DECIMAL PLACES and because the cube root of 64 is 4.

The problem now looks like this: (0.04)^1/2. The ½ instructs us to find the square root of 0.04. By rule number 2, the square root of 0.04 = 0.2. We were able to obtain this value because 0.04 has 2 DECIMAL PLACES and the square root of 4 is 2.

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Re: Which of the following is the value of root{3rd rt{0,000064} [#permalink]

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30 May 2016, 20:53
(3√0,000064)^(1/2) Or [(0.000064)^(1/3) ]^(1/2)

Start from the inner most value.

0.000064 = 64/1000000
(0.000064)^(1/3) = 4/100
{(0.000064)^(1/3) ]^(1/2) = (4/100)^(1/2) = 2/10

Correct Option: E
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Re: Which of the following is the value of root{3rd rt{0,000064} [#permalink]

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30 May 2016, 22:30
Bunuel wrote:
Which of the following is the value of $$\sqrt{\sqrt[3]{0.000064}}$$

(A) 0.004
(B) 0.008
(C) 0.02
(D) 0.04
(E) 0.2

We should attempt to write 0.000064 as a cube so that its cube root can be easily taken.

0.000064 = 64 * $$10^{-6}$$ = $$4^3$$ * $$10^{-6}$$

\sqrt[3]{0.000064} = 4 * $$10^{-2}$$ = $$2^2$$ * Square of ($$10^{-1}$$)

= 0.2

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Re: Which of the following is the value of root{3rd rt{0,000064} [#permalink]

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18 Jun 2016, 19:21
Here is another version..
Attachments

qustion.jpg [ 302.81 KiB | Viewed 540 times ]

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Re: Which of the following is the value of root{3rd rt{0,000064}   [#permalink] 18 Jun 2016, 19:21

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