How can I proof In a two column table that $\left(\frac{a}{b}\right)^n= \frac{a^n}{b^n}$ given $n$ is any positive integer.
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2Try using induction. – Umberto P. Oct 22 '19 at 01:11
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I need it in a two column proof – eniid15 Oct 22 '19 at 01:18
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I am afraid I have not used said "two column proofs" since geometry back in 8th grade...17 years ago? There should be nothing wrong with demonstrating your reasoning in paragraph form. – Math1000 Oct 22 '19 at 01:19
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Doesn't Distributive Property for Exponents work. Or have you not learned that? – Baker013273213 Oct 22 '19 at 01:39
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Presenting a "two-column" proof merely means justifying or explaining each step, which is a necessity in any proof, no matter its style. You can write a proof and then add spaces so they appear on the left side of the page. – DanielWainfleet Oct 22 '19 at 02:23
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Just use the laws of exponents. $(ab)^n=a^nb^n\Rightarrow (a/b)^n=(ab^{-1})^n=a^nb^{-n}=\frac{a^n}{b^n}$.
Simon Fraser
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To be precise, clearly $(a/b)^1 = a/b = a^1/b^1$ for $n=1$. Now assume that $(a/b)^n = a^n/b^n$ for some positive integer $n$. Then $$ (a/b)^{n+1} = (a/b)^n(a/b) = a^n/b^n\cdot a/b = a^{n+1}/b^{n+1}, $$ so we conclude by mathematical induction that $(a/b)^n = a^n/b^n$ for all positive integers $n$.
Math1000
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