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

Question

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 n integer.JPG

Bunuel · Accepted Answer

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 {n}*\sqrt {n} . Question: if  @(n)=0.5n  then  n=?

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

Answer: B.

KarishmaB · Answer

Just follow the rules of exponents. Answer will follow. 
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.

beyondgmatscore · Answer

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

asdlkj · Answer

n pow(1/3)* n pow(1/2)=0.5n

n pow (5/6)= 0.5n

n pow(1/6)=2

n= 64

sushantarora · Answer

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???????

Spidy001 · Answer

n^(5/6) = (1/2)n

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

=> n =0 or 64.

Answer is B.

gladvijay · Answer

50% of 729 is not an integer .. whereas @(729) is an integer

KarishmaB · Answer

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.

sushantarora · Answer

hi karishma,

seems like the best and easiest ans .. thank you so much .

erikvm · Answer

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

KarishmaB · Answer

Check out the blog posts on the link given in my signature below.

ScottTargetTestPrep · Answer

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

Answer: B

TheGuyDangerous · Answer

This can be written as (n)^1/3 * (n)^1/2 = 0.5n

= (n)^5/6 = 0.5n

= (n)^1/6 = 2

=> n = 2^6

=> n = 64

Ans B

QuantMadeEasy · Answer

@(n) = n^1/3 x n^1/2 = n^5/6

According to the condition 
n^5/6 = n/2
--> 2 = n^1/6
--> n =64

B is correct.

bumpbot · Answer

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

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