Power of powers.

$a={ b }^{ 2 }={ c }^{ 3 }={ d }^{ 5 }\\ \therefore b={ a }^{ 1/2 }\\ \therefore c={ a }^{ 1/3 }\\ \therefore d={ a }^{ 1/5 }\\ \therefore a,{ a }^{ 1/2 },{ a }^{ 1/3 },{ a }^{ 1/5 }\in Z\\ \therefore { a }^{ \frac { 1 }{ LCM(2,3,5) } }\in Z\\ \therefore { a }^{ 1/30 }\in Z\\ \\ So\quad we\quad can\quad say\quad that\quad a={ z }^{ 30 },\quad where\quad z\in Z\\ \min _{ { z }^{ 5 }\neq { z }^{ 3 }\neq { z }^{ 2 }\neq z }{ z } =2\\ \therefore \min { a } ={ 2 }^{ 30 }=1073741824$

We know that $a, \sqrt { a } , \sqrt [ 3 ]{ a },$ and $\sqrt [ 5 ]{ a }$ must be positive integers. Another way of looking at this is that ${ a }^{ \frac { 1 }{ 2 } }, { a }^{ \frac { 1 }{ 3 } },$ and ${ a }^{ \frac { 1 }{ 5 } }$ must have positive integer exponents. This is only achieved if the exponent of $a$ is a multiple of 2, 3, and 5. Since we're looking for the smallest $a$ , it's natural that we want the $lcm(2,3,5)=30$ to be the exponent.

So we know that $a={ x }^{ 30 }$ . Again, since we want the smallest integer $a$ , we have to pick $a={ 2 }^{ 30 }$ , since $1$ wouldn't give us distinct integers.

This gives us ${ 2 }^{ 30 }=1073741824$ as the smallest positive $a$ .

Actually with the given conditions , 2^(lcm(2,3,5)) will be the answer...

We have $lcm\left( 2,3,5 \right) =30$

Now we know that $(a={ x }^{ 30 },$ for some $x\in N$

So we have solutions for $a$ as ${ 1 }^{ 30 },{ 2 }^{ 30 },{ 3 }^{ 30 },{ 4 }^{ 30 }.....$

But ${ 2 }^{ 30 }$ is number within our conditions.

So our answer is $1073741824$ .

In the prime factorization of a, all the exponents must be divisible by $2, 3,$ and $5$ .

$lcm(2, 3, 5) = 30$

Obviously, we want to raise one prime to the $30th$ power.

The least prime is $2$ .

$2 ^ {30} = 1073741824$

Since all are distinct, it follows that a > 1. a. b, c, d all have a unique prime factorization.. If a is the 2nd ,3rd, and 5th power of b, c, and d, the powers of its prime factors will be divisible by 2,3, and 5 based on the unique prime factorization of b, c and d. The least common multiple of 2,3,5 is 30. The smallest prime is 2. Thus $2^{30} = 1,073,741,824$ is the least possible value of a.

2^2x3x5 =2^30