How Do I Simplify $1 - \frac{1}{2^{m-1}} + \frac{1}{2^{m}}$

Question

Someone help. Somehow my textbook says that it simplifies to $1 - \frac{2}{2^{m}} + \frac{1}{2^{m}}$ I don't see this at all.

Answer 1

hint: use the property of exponents: $2^{m-1} = \dfrac{2^m}{2}$, and note that: $\dfrac{1}{\dfrac{2^m}{2}} = \dfrac{2}{2^m}$

Answer 2

They have just multiplied the second term by 1 but in a special way:

$$ \begin{split} &1-\frac{1}{2^{m-1}}+\frac{1}{2^m} \\ =&1-\frac{1}{2^{m-1}}\cdot \frac{2}{2}+\frac{1}{2^m} \\=&1-\frac{2}{2^m}+\frac{1}{2^m} \end{split} $$

Answer 3

$$1 - (\frac{1}{2})^{m-1} + (\frac{1}{2})^{m}$$ $$\Rightarrow 1 - \frac{1^{m-1}}{2^{m-1}} + \frac{1^{m}}{2^{m}}$$ $$\Rightarrow 1 - \frac{1}{2^{m-1}} + \frac{1}{2^{m}}$$ $$\Rightarrow 1 - \frac{1\cdot2}{2^{m-1}2} + \frac{1}{2^{m}}$$ $$\Rightarrow 1 - \frac{2}{2^{m}} + \frac{1}{2^{m}}$$

Answer 4

One property of exponents is that $x^{m + n} = x^{m}x^{n}$, and another one is $x^{-m} = \frac{1}{x^{m}}$. In your case, it's the second term which has been simplified, the rest is unchanged. $\frac{1}{2^{m-1}} = \frac{1}{2^{m}2^{-1}}$. Using the second property I stated in reverse will give you $\frac{1}{2^{m}2^{-1}} = \frac{2^{1}}{2^{m}} = \frac{2}{2^{m}}$. I hope this helped.