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# If @(n) is defined as the product of the cube root of n and

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If @(n) is defined as the product of the cube root of n and  [#permalink]

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Updated on: 10 Feb 2014, 01:50
1
19
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Difficulty:

45% (medium)

Question Stats:

70% (02:05) correct 30% (02:28) wrong based on 672 sessions

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If @(n) is defined as the product of the cube root of n and the positive square root of n, then for what number n does @(n)=50 percent of n?

A. 16
B. 64
C. 100
D. 144
E. 729

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The proof of understanding is the ability to explain it.

Originally posted by GMATD11 on 28 Feb 2011, 03:12.
Last edited by Bunuel on 10 Feb 2014, 01:50, edited 1 time in total.
Renamed the topic and edited the question.
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28 Feb 2011, 03:27
9
10
If @(n) is defined as the product of the cube root of n and the positive square root of n, then for what number n does @(n)=50 percent of n?

A. 16
B. 64
C. 100
D. 144
E. 729

Given: $$@(n)=\sqrt[3]{n}*\sqrt[2]{n}$$. Question: if $$@(n)=0.5n$$ then $$n=?$$

So we have that $$\sqrt[3]{n}*\sqrt[2]{n}=\frac{1}{2}*n$$ --> $$2*\sqrt[3]{n}*\sqrt[2]{n}=n$$ --> take to the 6th power --> $$64*n^2*n^3=n^6$$ --> $$n=64$$.

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Manager
Joined: 14 Feb 2011
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28 Feb 2011, 03:19
for 16, cube root is 2* cube root of 2 and positive square root is 4, so @n = 2*2^(1/3)*4 = 8 *2(1/3) so greater than 8 or greater than 0.5n

Similarly for 64, it is 4*8 = 32 = 0.5*64, so B is correct.

For E, the cube root is 9 and positive square root is 27, so 27*9 is not equal to 0.5*729, so incorrect
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Joined: 28 Feb 2011
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28 Feb 2011, 07:20
n pow(1/3)* n pow(1/2)=0.5n

n pow (5/6)= 0.5n

n pow(1/6)=2

n= 64
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28 Feb 2011, 19:51
3
5
GMATD11 wrote:
From B and E whts wrong with E

Cube root is the power of 1/3. Square root is the power of 1/2

$$n^{\frac{1}{3}}*n^{\frac{1}{2}} = \frac{n}{2}$$
You need to find n. So bring all n's together on one side of the equation and everything else on the other side.

Adding the exponents, $$n^{\frac{5}{6}} = \frac{n}{2}$$
Clubbing n's together, $$2 = n^{1-\frac{5}{6}}$$
$$n = 2^6 = 64$$

Hence it cannot be 729. If in the question, rather than half, we had a third, answer would have been 729.
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01 Aug 2011, 09:44
the cube root of what integer power of 2 is closest to 50?

1)16 2) 17 3)18 4 ) 19 5) 20

can u pls help me in this by a quicker solution???????
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Joined: 01 Feb 2011
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01 Aug 2011, 11:50
n^(5/6) = (1/2)n

=> n^6-2^6*n^5 = 0

=> n =0 or 64.

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01 Aug 2011, 13:43
1
GMATD11 wrote:
From B and E whts wrong with E

50% of 729 is not an integer .. whereas @(729) is an integer
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01 Aug 2011, 21:08
1
sushantarora wrote:
the cube root of what integer power of 2 is closest to 50?

1)16 2) 17 3)18 4 ) 19 5) 20

can u pls help me in this by a quicker solution???????

Look at the powers of 2.

2^1 = 2
2^2 = 4
2^3 = 8
2^4 = 16
2^5 = 32
2^6 = 64
2^7 = 128
Since it is an exponential increase, the result increases much more as you go to higher and higher powers.
Which powers of 2 are around 50?
2^5 = 32
2^6 = 64
50 is almost in the middle of the two of them but closer to 64. Also, the result increases more with higher powers so I would expect 50 to be almost 2^(5.6) or a little higher.

If you find the cube root of 2^18, you will get (2^18)^(1/3) = 2^6
If you find the cube root of 2^17, you will get (2^17)^(1/3) = 2^(5.667)
This is the closest. Answer is 17.
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01 Aug 2011, 23:35
hi karishma,

seems like the best and easiest ans .. thank you so much .
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Re: If (n) is defined as the product of the cube root of n and  [#permalink]

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10 Mar 2015, 03:31
VeritasPrepKarishma wrote:
GMATD11 wrote:
From B and E whts wrong with E

Cube root is the power of 1/3. Square root is the power of 1/2

$$n^{\frac{1}{3}}*n^{\frac{1}{2}} = \frac{n}{2}$$
You need to find n. So bring all n's together on one side of the equation and everything else on the other side.

Adding the exponents, $$n^{\frac{5}{6}} = \frac{n}{2}$$
Clubbing n's together, $$2 = n^{1-\frac{5}{6}}$$
$$n = 2^6 = 64$$

Hence it cannot be 729. If in the question, rather than half, we had a third, answer would have been 729.

I really need to study these formulas, where do I suggest I go?
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Re: If (n) is defined as the product of the cube root of n and  [#permalink]

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10 Mar 2015, 06:05
erikvm wrote:
VeritasPrepKarishma wrote:
GMATD11 wrote:
From B and E whts wrong with E

Cube root is the power of 1/3. Square root is the power of 1/2

$$n^{\frac{1}{3}}*n^{\frac{1}{2}} = \frac{n}{2}$$
You need to find n. So bring all n's together on one side of the equation and everything else on the other side.

Adding the exponents, $$n^{\frac{5}{6}} = \frac{n}{2}$$
Clubbing n's together, $$2 = n^{1-\frac{5}{6}}$$
$$n = 2^6 = 64$$

Hence it cannot be 729. If in the question, rather than half, we had a third, answer would have been 729.

I really need to study these formulas, where do I suggest I go?

Here are the basics of exponents and roots:

http://www.veritasprep.com/blog/2011/07 ... eparation/
http://www.veritasprep.com/blog/2011/07 ... ration-ii/
http://www.veritasprep.com/blog/2011/07 ... s-applied/
http://www.veritasprep.com/blog/2011/08 ... -the-gmat/
http://www.veritasprep.com/blog/2011/08 ... exponents/
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Re: If (n) is defined as the product of the cube root of n and  [#permalink]

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28 Jun 2017, 17:01
GMATD11 wrote:
If @(n) is defined as the product of the cube root of n and the positive square root of n, then for what number n does @(n)=50 percent of n?

A. 16
B. 64
C. 100
D. 144
E. 729

We are given that @(n) is defined as the product of the cube root of n and the positive square root of n.

We need to determine for what number n does @(n) = 50 percent of n.

We see that @n = (∛n)(√n) = (n^(⅓))(n^(½)) = n^(⅓ + ½) = n^(⅚)

We need to determine a value for n when:

n^(⅚) = 0.5n

2n^(⅚) = n

2 = n/n^(⅚)

2 = n^(⅙)

2^6 = n

n = 64

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Re: If @(n) is defined as the product of the cube root of n and  [#permalink]

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19 Sep 2018, 09:22
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Re: If @(n) is defined as the product of the cube root of n and   [#permalink] 19 Sep 2018, 09:22
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