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I am trying to Simplify : $$\frac{ (2^8 + 6^8)}{(9^8 + 3^8)}$$

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5 Answers 5

up vote 5 down vote accepted

$$\frac{2^8+6^8}{3^8+9^8}=\frac{2^8+(2^8\times 3^8)}{3^8+(3^8\times 3^8)}=\frac{2^8(1+3^8)}{3^8(1+3^8)}=\frac{2^8}{3^8}=\left(\frac{2}{3}\right)^8$$ Note that whenever you want to remove some terms from denominator and numerator of a fraction, those terms should be not zero. And here $1+3^8\neq 0$ obviously.

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Well done...the thought process speaks for itself +1 –  amWhy Feb 9 '13 at 0:07
    
@amWhy: Please reopen yours. It contains some points which I didn't even note them. –  B. S. Feb 9 '13 at 5:31
    
Okay, thanks. Reopened ;-) –  amWhy Feb 9 '13 at 14:34

$$\frac{2^8+6^8}{9^8+3^8} = \frac{2^8(1+3^8)}{3^8(1+3^8)} = \frac{2^8}{3^8} = \left(\frac{2}{3}\right)^8$$

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$$ \begin{aligned}= & \dfrac{2^8(1 + 3^8)}{3^8(1 + 3^8)} \\ \\ \\ = & \frac{2^8}{3^8} \\ \\ \\ =& \left(\frac{2}{3}\right)^8\end{aligned}$$Factoring and cancelling is the best way which crosses one's mind in such cases.

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$$\frac{ 2^8 + 6^8}{9^8 + 3^8}=\frac{ 2^8 + (2\cdot3)^8}{(3\cdot3)^8 + 3^8}=\frac{ 2^8 + 2^8\cdot3^8}{3^8\cdot3^8 + 3^8}=\frac{ 2^8(1 + 3^8)}{3^8(1 + 3^8)}=\frac{ 2^8}{3^8}=\Big(\frac{ 2}{3}\Big)^8$$

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Just to elaborate on the above answers, using the power rules:

$$(ab)^n = a^n\cdot b^n,\quad \left(\frac{a^n}{b^n}\right) = \left(\frac ab\right)^n$$

We work on simplifying your expression:

$$\frac{ (2^8 + 6^8)}{(9^8 + 3^8)} = \frac{(2^8\cdot 1^8 + 2^8\cdot 3^8)}{(3^8 \cdot 3^8 + 3^8\cdot 1^8)}$$

Factoring out common terms gives us $$\frac{ 2^8(1 + 3^8)}{3^8(3^8 + 1)} = \frac{ 2^8(1 + 3^8)}{3^8(1 + 3^8)}$$

Canceling common factors from numerator and denominator: $$\frac{ 2^8}{3^8} = \left(\frac23\right)^8$$

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