Solution
- a and b are integers. a | b means that integer a can divide integer b with no remainders.
- Let the Quotient of the division be k, so we can say:
[tex]\begin{gathered} a|b=k \\ \\ \text{ Put in an easier way, we have:} \\ \frac{b}{a}=k \\ \\ where, \\ k\text{ is an integer since }a\text{ directly divides b} \end{gathered}[/tex]- Now, we are asked to find
[tex]a^n|b^n[/tex]- Again, we can rewrite this as:
[tex]\frac{b^n}{a^n}[/tex]- We can rewrite this expression using the law of exponents that says:
[tex]\frac{x^m}{y^m}=(\frac{x}{y})^m[/tex]- Applying this law, we have:
[tex]\frac{b^n}{a^n}=(\frac{b}{a})^n[/tex]- But we already know that
[tex]\frac{b}{a}=k[/tex]- Thus, we have that:
[tex]\begin{gathered} \frac{b^n}{a^n}=k^n \\ \\ That\text{ is,} \\ a^n|b^n=k^n \\ for\text{ all positive integers n} \\ \\ k^n\text{ is an integer as well because }k\text{ is an integer.} \end{gathered}[/tex]- Therefore, we have successfully proved the assertion