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

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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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

Answer is E

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

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?

Thanks for your help!

Yes, your reasoning is correct.

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

Answer = E

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

Attached is a visual that should help.

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.

Answer E.

(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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