SAT question about integers and exponents

Question

If $a$ and $b$ are positive integers and $$(a^\frac{1}{2}\times b^\frac{1}{3})^6=432$$ What is the value of $ab$?

Answer 1

Note that $$432 = 2^4\cdot 3^3 = 3^3\cdot4^2$$ and the LHS can be written as $$(a^\frac{1}{2} \times b^\frac{1}{3})^6 = a^3\cdot b^2$$ So $a = 3, b = 4$ $\Rightarrow ab = 12$.

Answer 2

As a hint, I'd suggest:

• Write out $(a^{\frac{1}{2}}\times b^{\frac{1}{3}})^3$,
• Factorize $432$ into prime numbers.

Answer 3

the relation is equivalent to $a^3.b^2=432$ now prime factorizing we get $432=2^4.3^3=4^2.3^3$ thus $a=3,b=4$ thus $ab=12$