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If $a$ is an integer and $b$ is a fourth power of an integer such that $ab$ is the fourth power of an integer, explain why $a$ is also a fourth power of an integer.

Show that $a$ is the fourth power of an integer if: $$b^4$$ $$(ab)^4$$

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HINT: What can you say about the exponents in the prime decomposition of a fourth power?

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Would the prime decomposition of a fourth power be $p^4=p \cdot p \cdot p \cdot p$. –  Dmitri.Mendeleev Dec 10 '12 at 4:58
    
@Dmitri: Not necessarily: not all fourth powers are fourth powers of a prime. The prime decomposition of $6^4$ is $2^4\cdot3^4$. The prime decomposition of $12^4$ is $2^8\cdot3^4$. –  Brian M. Scott Dec 10 '12 at 5:00
    
I see. How does this help me show that $a$ is to the fourth power? –  Dmitri.Mendeleev Dec 10 '12 at 5:01
    
@Dmitri: Can you yet answer the question in my hint? The answer is the key to showing that $a$ is a fourth power. –  Brian M. Scott Dec 10 '12 at 5:04
    
The exponents are a multiple of $4$? –  Dmitri.Mendeleev Dec 10 '12 at 5:06

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