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# help

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If $x=735$ and $ax$ is a perfect square where $a$ is a positive integer, what is the smallest possible value of $\sqrt{ax}$?

Mar 1, 2022

#1
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$$x=735$$

$$735a=perfect$$ $$square$$

well, $$735=7\times105=7\times5\times3\times7$$

$$735=3\times5\times7^2$$

in order to get a perfect square, you will need a 3 and a 5 because every number in the prime factorization needs to have a square.

$$\sqrt{735\cdot15}=\sqrt{3^2\cdot5^2\cdot7^2}=3\cdot5\cdot7=105$$

notice that you can take a shortcut by multiplying 3x5x7.

I'll let you wonder why, but it works all the time!

Mar 1, 2022

#1
+4

$$x=735$$

$$735a=perfect$$ $$square$$

well, $$735=7\times105=7\times5\times3\times7$$

$$735=3\times5\times7^2$$

in order to get a perfect square, you will need a 3 and a 5 because every number in the prime factorization needs to have a square.

$$\sqrt{735\cdot15}=\sqrt{3^2\cdot5^2\cdot7^2}=3\cdot5\cdot7=105$$

notice that you can take a shortcut by multiplying 3x5x7.

I'll let you wonder why, but it works all the time!

XxmathguyxX Mar 1, 2022