# Thread: Simplify this equation w/ fractional expnents

1. ## Simplify this equation w/ fractional expnents

I'm working on an induction problem, but can't show my work for simplifying this equation:

1 - (1/2)^k + 1/2^(k+1) --------> Simplify ---------> 1 - (1/2)^(k + 1)

I'm just assuming that's what it simplifys to. Can someone show me the work for this with the right answer if it's wrong.

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2. $\displaystyle 1-\left (\left (\frac{1}{2} \right )^{k}-\left (\frac{1}{2} \right )^{k+1} \right )=1-\left ( \frac{1}{2} \right )^{k+1}$,you simplified right.

3. $\displaystyle 1-\left (\left (\frac{1}{2} \right )^{k}-\left (\frac{1}{2} \right )^{k+1} \right )$$\displaystyle =1- \left (\left (\frac{1}{2} \right )^{k}-\left (\frac{1}{2} \right )^k\left ( \frac{1}{2} \right ) \right )$$\displaystyle =1-\left ( \frac{1}{2} \right )^k\left ( 1-\frac{1}{2} \right )$$\displaystyle =1-\left ( \frac{1}{2} \right )\left ( \frac{1}{2} \right )^k$$\displaystyle =1-\left ( \frac{1}{2} \right )^{k+1}$

4. Hello everyone

Alternatively you might find it easier to say
$\displaystyle 1-\left(\frac12\right)^k+\left(\frac12\right)^{k+1}$
$\displaystyle =1-2\times\left(\frac12\right)^{k+1}+1\times\left(\fr ac12\right)^{k+1}$

$\displaystyle =1-\left(\frac12\right)^{k+1}$